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<b>let </b><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>let </b><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>;<br/><b>let </b><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>;<br/>







<b>assume </b><a NAME="E1:75"/><i><font color="Green" title="E44">A1</font></i>: 
( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> &amp; <font color="Maroon" title="c3">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 ;<br/>

<a NAME="E2:75"/><b>then </b><i><font color="Green" title="E45">A2</font></i>: 
<font color="Maroon" title="c1">G1</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 
<b>by </b><i><a class="txt" href="glib_002.html#E24"><i><font color="Green" title="E23">Lm12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">G1A</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c3">v1</font>;<br/>
<a NAME="E4:75"/><i><font color="Green" title="E46">A3</font></i>: 
<font color="Maroon" title="c2">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c5">G1A</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="glib_002.html#E2:75"><i><font color="Green" title="E45">A2</font></i></a>, <a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">Def10</a></i>;<br/>
<a NAME="E5:75"/><i><font color="Green" title="E47">A4</font></i>: 
not <font color="Maroon" title="c1">G1</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_002.html#T38" target="_self" title="GLIB_002:th.38">Th38</a></i>;<br/>
<b>then </b><i><font color="Green" title="E48">A5</font></i>:  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c5">G1A</font> = 
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
;<br/>
<i><font color="Green" title="E49">A6</font></i>:  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">G1A</font> = 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_002.html#E5:75"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>
<br/>.= 
<font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>
<br/>.= 
<font color="Maroon" title="c2">G2</font> <a href="glib_000.html#K22" title="GLIB_000:func.22">.edgesBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">v2</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_000.html#T93" title="GLIB_000:th.93">GLIB_000:93</a></i>
;<br/>
<b>let </b><font color="Maroon" title="c6">G2A</font> be   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">v2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="glib_002.html#D10" target="_self" title="GLIB_002:def.10">GLIB_002:def 10</a><br/>

<a NAME="E8:75"/><i><font color="Green" title="E50">A7</font></i>: 
not <font color="Maroon" title="c2">G2</font> is <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="txt" href="glib_002.html#E5:75"><i><font color="Green" title="E47">A4</font></i></a>, <a class="ref" href="glib_000.html#T92" title="GLIB_000:th.92">GLIB_000:92</a></i>;<br/>
<a NAME="E9:75"/><b>then </b><i><font color="Green" title="E51">A8</font></i>: 
 <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c6">G2A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c5">G1A</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E6:75"><i><font color="Green" title="E48">A5</font></i></a>, <a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/>
<a NAME="E10:75"/><i><font color="Green" title="E52">A9</font></i>: 
 <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c6">G2A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c5">G1A</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E7:75"><i><font color="Green" title="E49">A6</font></i></a>, <a class="txt" href="glib_002.html#E8:75"><i><font color="Green" title="E50">A7</font></i></a>, <a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/>
<a NAME="E11:75"/>
<font color="Maroon" title="c2">G2</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="txt" href="glib_002.html#E1:75"><i><font color="Green" title="E44">A1</font></i></a>, <a class="ref" href="glib_000.html#T90" title="GLIB_000:th.90">GLIB_000:90</a></i>;<br/>
<a NAME="E12:75"/><b>then </b>
<font color="Maroon" title="c6">G2A</font> is    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font>
 
<b>by </b><i><a class="ref" href="glib_000.html#T46" title="GLIB_000:th.46">GLIB_000:46</a></i>;<br/>
<b>hence </b><a NAME="E13:75"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a>  <a href="card_1.html#NR2" title="CARD_1:NR.2">&lt;`</a> <font color="Maroon" title="c6">G2A</font> <a href="glib_002.html#K4" title="GLIB_002:func.4">.numComponents()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E4:75"><i><font color="Green" title="E46">A3</font></i></a>, <a class="txt" href="glib_002.html#E9:75"><i><font color="Green" title="E51">A8</font></i></a>, <a class="txt" href="glib_002.html#E10:75"><i><font color="Green" title="E52">A9</font></i></a>, <a class="txt" href="glib_002.html#E24"><i><font color="Green" title="E23">Lm12</font></i></a>, <a class="ref" href="glib_000.html#T89" title="GLIB_000:th.89">GLIB_000:89</a></i>;<br/>


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