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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[ <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>] <b>means </b>( $1 is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a> implies $1 <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default">$1 <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 );<br/>
<a NAME="E1:76"/><i><font color="Green" title="E45">A1</font></i>: 
for <font color="Olive" title="b1">G</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>]
 
<b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/>
<div><i><font color="Green" title="E46">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:76_1"/><i><font color="Green" title="E47">A3</font></i>: 
for <font color="Olive" title="b1">Gk</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>  st <font color="Olive" title="b1">Gk</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">Gk</font>]
 ;<br/><b>let </b><font color="Maroon" title="c2">Gk1</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>;<br/><b>assume </b><a NAME="E2:76_1"/><i><font color="Green" title="E48">A4</font></i>: 
<font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1"/><i><font color="Green" title="E49">A5</font></i>: 
<font color="Maroon" title="c2">Gk1</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="76_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:76_1_1_1"/>
<font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><a NAME="E2:76_1_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:76_1"><i><font color="Green" title="E48">A4</font></i></a></i>;<br/><b>hence </b><a NAME="E3:76_1_1_1"/>
contradiction
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c3">Gk1'</font> = <font color="Maroon" title="c2">Gk1</font> as  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> non <a href="glib_000.html#V4" title="GLIB_000:attr.4">trivial</a> <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> <b>by </b><i><a class="txt" href="glib_002.html#E1:76_1_1"><i><font color="Green" title="E49">A5</font></i></a>, <a class="ref" href="glib_000.html#T29" title="GLIB_000:th.29">GLIB_000:29</a></i>;<br/><b>consider </b><font color="Maroon" title="c4">v1</font>, <font color="Maroon" title="c5">v2</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">Gk1'</font><b> such that </b><br/><a NAME="E5:76_1_1"/><i><font color="Green" title="E50">A6</font></i>: 
( <font color="Maroon" title="c4">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">v2</font> &amp; not <font color="Maroon" title="c4">v1</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> &amp; not <font color="Maroon" title="c5">v2</font> is <a href="glib_002.html#V5" title="GLIB_002:attr.5">cut-vertex</a> )
 <b>by </b><i><a class="txt" href="glib_002.html#E33"><i><font color="Green" title="E32">Lm21</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">Gkb</font> being   <a href="glib_000.html#NM6" title="GLIB_000:NM.6">removeVertex</a> of <font color="Maroon" title="c3">Gk1'</font>,<font color="Maroon" title="c4">v1</font>;<br/><a NAME="E7:76_1_1"/><i><font color="Green" title="E51">A7</font></i>: 
<font color="Maroon" title="c6">Gkb</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#E5:76_1_1"><i><font color="Green" title="E50">A6</font></i></a>, <a class="txt" href="glib_002.html#E32"><i><font color="Green" title="E31">Lm20</font></i></a></i>;<br/><a NAME="E8:76_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c6">Gkb</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c6">Gkb</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a>  )
 <b>by </b><i><a class="txt" href="glib_002.html#E2:76_1"><i><font color="Green" title="E48">A4</font></i></a>, <a class="ref" href="glib_000.html#T51" title="GLIB_000:th.51">GLIB_000:51</a></i>;<br/><a NAME="E9:76_1_1"/><b>then </b><i><font color="Green" title="E52">A8</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E1:76_1"><i><font color="Green" title="E47">A3</font></i></a>, <a class="txt" href="glib_002.html#E7:76_1_1"><i><font color="Green" title="E51">A7</font></i></a></i>;<br/><a NAME="E10:76_1_1"/>
not <font color="Maroon" title="c4">v1</font> is <a href="glib_000.html#V10" title="GLIB_000:attr.10">isolated</a>
 <b>by </b><i><a class="ref" href="glib_002.html#T2" target="_self" title="GLIB_002:th.2">Th2</a></i>;<br/><a NAME="E11:76_1_1"/><b>then </b>
<font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D51" title="GLIB_000:def.51">GLIB_000:def 51</a></i>;<br/><a NAME="E12:76_1_1"/><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="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="card_2.html#T59" title="CARD_2:th.59">CARD_2:59</a></i>;<br/><a NAME="E13:76_1_1"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T3" title="NAT_1:th.3">NAT_1:3</a></i>;<br/><a NAME="E14:76_1_1"/><b>then </b>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E15:76_1_1"/><b>then </b>
<font color="Maroon" title="c1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p2">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">v1</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E9:76_1_1"><i><font color="Green" title="E52">A8</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>;<br/><b>hence </b><a NAME="E16:76_1_1"/>
<font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">Gk1</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E2:76_1"><i><font color="Green" title="E48">A4</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E4:76_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">Gk1</font>]
 ;<br/></div><b>end;</b></div>
<a NAME="E3:76"/><i><font color="Green" title="E47">A9</font></i>: 
for <font color="Olive" title="b1">G</font> being <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">G</font>]
 
<b>from </b><i><a class="ref" href="glib_000.html#S1" title="GLIB_000:sch.1">GLIB_000:sch 1</a>(<a class="txt" href="glib_002.html#E1:76"><i><font color="Green" title="E45">A1</font></i></a>, <a class="txt" href="glib_002.html#E2:76"><i><font color="Green" title="E46">A2</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <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>;<br/>

<b>thus </b><a NAME="E4:76"/>
<font color="Maroon" title="c1">G</font> <a href="glib_000.html#K16" title="GLIB_000:func.16">.order()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K18" title="GLIB_000:func.18">.size()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E3:76"><i><font color="Green" title="E47">A9</font></i></a></i>;<br/>


</div>
