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<b>let </b><font color="Maroon" title="c1">G1</font> be  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>;<br/><b>let </b><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">G1</font>;<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">G1</font>,<font color="Maroon" title="c2">v</font>;<br/>





<b>assume </b><a NAME="E1:33"/><i><font color="Green" title="E17">A1</font></i>: 
<font color="Maroon" title="c2">v</font> is <a href="glib_000.html#V11" title="GLIB_000:attr.11">endvertex</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">ev</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:33"/><i><font color="Green" title="E18">A2</font></i>: 
( <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c4">ev</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; not <font color="Maroon" title="c4">ev</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="ref" href="glib_000.html#D53" title="GLIB_000:def.53">GLIB_000:def 53</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">VG</font> =  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">VG2</font> =  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">v1'</font>, <font color="Maroon" title="c8">v2'</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c3">G2</font>;<br/><b>reconsider </b><font color="Maroon" title="c9">v1</font> = <font color="Maroon" title="c7">v1'</font>, <font color="Maroon" title="c10">v2</font> = <font color="Maroon" title="c8">v2'</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="ref" href="glib_000.html#T45" title="GLIB_000:th.45">GLIB_000:45</a></i>;<br/><b>consider </b><font color="Maroon" title="c11">W</font> being    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:33_1"/><i><font color="Green" title="E19">A3</font></i>: 
<font color="Maroon" title="c11">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/><a NAME="E4:33_1"/>
( <font color="Maroon" title="c7">v1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> &amp; <font color="Maroon" title="c8">v2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font> )
 ;<br/><a NAME="E5:33_1"/><b>then </b>
( <font color="Maroon" title="c7">v1'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; <font color="Maroon" title="c8">v2'</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/><a NAME="E6:33_1"/><b>then </b><i><font color="Green" title="E20">A4</font></i>: 
( not <font color="Maroon" title="c9">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c10">v2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E7:33_1"/><b>then </b><i><font color="Green" title="E21">A5</font></i>: 
( <font color="Maroon" title="c9">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> &amp; <font color="Maroon" title="c10">v2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">v</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>consider </b><font color="Maroon" title="c12">T</font> being   <a href="glib_001.html#NM10" title="GLIB_001:NM.10">Trail</a> of <font color="Maroon" title="c11">W</font>;<br/><a NAME="E9:33_1"/><i><font color="Green" title="E22">A6</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c9">v1</font>,<font color="Maroon" title="c10">v2</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1"><i><font color="Green" title="E19">A3</font></i></a>, <a class="ref" href="glib_001.html#T161" title="GLIB_001:th.161">GLIB_001:161</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_1"/><i><font color="Green" title="E23">A7</font></i>: 
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 ;<br/><a NAME="E2:33_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  or <font color="Maroon" title="c2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  )
 <b>by </b><i><a class="txt" href="glib_002.html#E1:33"><i><font color="Green" title="E17">A1</font></i></a>, <a class="txt" href="glib_002.html#E1:33_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="glib_001.html#T144" title="GLIB_001:th.144">GLIB_001:144</a></i>;<br/><b>hence </b><a NAME="E3:33_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E7:33_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="glib_002.html#E9:33_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/></div><b>end;</b></div><a NAME="E3:33_1_1"/><b>then </b>
not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E4:33_1_1"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">v</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#E1:33_1_1"><i><font color="Green" title="E23">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E5:33_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/></div><b>end;</b></div><a NAME="E11:33_1"/><b>then </b><i><font color="Green" title="E23">A8</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K25" title="GLIB_000:func.25">the_Vertices_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E12:33_1"/><i><font color="Green" title="E24">A9</font></i>: 
( <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v1</font> &amp; <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">v2</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E9:33_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c13">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:33_1_2"/><i><font color="Green" title="E25">A10</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c12">T</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a> 
 ;<br/><b>then consider </b><font color="Maroon" title="c14">n</font> being  <a href="abian.html#V1" title="ABIAN:attr.1">even</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> <b> such that </b><br/><a NAME="E3:33_1_2"/><i><font color="Green" title="E26">A11</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">n</font> &amp; <font color="Maroon" title="c14">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font> &amp; <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">e</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T100" title="GLIB_001:th.100">GLIB_001:100</a></i>;<br/><a NAME="E4:33_1_2"/>
<font color="Maroon" title="c14">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">T</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><b>then consider </b><font color="Maroon" title="c15">n1</font> being  <a href="abian.html#NV2" title="ABIAN:NV.2">odd</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> <b> such that </b><br/><a NAME="E6:33_1_2"/><i><font color="Green" title="E27">A12</font></i>: 
( <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 &amp; <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">T</font> &amp; <font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c12">T</font> &amp; <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font>,<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>,<font color="Maroon" title="c1">G1</font> )
 <b>by </b><i><a class="ref" href="glib_001.html#T10" title="GLIB_001:th.10">GLIB_001:10</a></i>;<br/><a NAME="E7:33_1_2"/><i><font color="Green" title="E28">A13</font></i>: 
( <font color="Maroon" title="c15">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font> &amp; <font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font> )
 <b>by </b><i><a class="txt" href="glib_002.html#E6:33_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:33_1_2_1"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 ;<br/><a NAME="E2:33_1_2_1"/><b>then </b><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">ev</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33"><i><font color="Green" title="E18">A2</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="33_1_2_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:33_1_2_1_1_1"/>
( <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> or <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> )
 </span><b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="glib_002.html#E6:33_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="glib_002.html#E1:33_1_2_1"><i><font color="Green" title="E29">A14</font></i></a>, <a class="ref" href="glib_000.html#T68" title="GLIB_000:th.68">GLIB_000:68</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_1"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_1"/><i><font color="Green" title="E31">A16</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E32">A17</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <font color="Maroon" title="c15">n1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:33_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="glib_001.html#D8" title="GLIB_001:def.8">GLIB_001:def 8</a></i>;<br/><a NAME="E3:33_1_2_1_1_1_1"/><i><font color="Green" title="E33">A18</font></i>: 
<font color="Maroon" title="c15">n1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 <b>by </b><i><a class="txt" href="glib_002.html#E6:33_1"><i><font color="Green" title="E20">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a>, <a class="txt" href="glib_002.html#E12:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="glib_002.html#E1:33_1_2_1_1_1_1"><i><font color="Green" title="E31">A16</font></i></a></i>;<br/><a NAME="E4:33_1_2_1_1_1_1"/><i><font color="Green" title="E34">A19</font></i>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n1</font>
 <b>by </b><i><a class="ref" href="heyting3.html#T1" title="HEYTING3:th.1">HEYTING3:1</a></i>;<br/><a NAME="E5:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E35">A20</font></i>: 
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">n1</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2_1_1_1_1"><i><font color="Green" title="E33">A18</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c16">n2</font> = <font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 as  <a href="abian.html#V1" title="ABIAN:attr.1">even</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>  <b>by </b><i><a class="txt" href="glib_002.html#E4:33_1_2_1_1_1_1"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="int_1.html#T18" title="INT_1:th.18">INT_1:18</a></i>;<br/><a NAME="E7:33_1_2_1_1_1_1"/>
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E7:33_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="glib_002.html#E2:33_1_2_1_1_1_1"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="glib_002.html#E5:33_1_2_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="glib_001.html#T12" title="GLIB_001:th.12">GLIB_001:12</a></i>;<br/><a NAME="E8:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E36">A21</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="glib_002.html#E2:33_1_2_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:33_1_2_1_1_1_1"/>
<font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E10:33_1_2_1_1_1_1"/><b>then </b><i><font color="Green" title="E37">A22</font></i>: 
<font color="Maroon" title="c15">n1</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 0
 <b>by </b><i><a class="txt" href="glib_002.html#E6:33_1_2"><i><font color="Green" title="E27">A12</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><a NAME="E11:33_1_2_1_1_1_1"/>
1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">n1</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E5:33_1_2_1_1_1_1"><i><font color="Green" title="E35">A20</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><a NAME="E12:33_1_2_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c16">n2</font>
 <b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E13:33_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="glib_002.html#E8:33_1_2_1_1_1_1"><i><font color="Green" title="E36">A21</font></i></a>, <a class="txt" href="glib_002.html#E10:33_1_2_1_1_1_1"><i><font color="Green" title="E37">A22</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="33_1_2_1_1_1_2"><b>suppose </b></a><a NAME="E1:33_1_2_1_1_1_2"/><i><font color="Green" title="E31">A23</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 ;<br/><div class="add"><a NAME="E2:33_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E32">A24</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="glib_001.html#K5" title="GLIB_001:func.5">.vertexAt</a> <span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:33_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="glib_001.html#D8" title="GLIB_001:def.8">GLIB_001:def 8</a></i>;<br/><a NAME="E3:33_1_2_1_1_1_2"/><i><font color="Green" title="E33">A25</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E7:33_1"><i><font color="Green" title="E21">A5</font></i></a>, <a class="txt" href="glib_002.html#E12:33_1"><i><font color="Green" title="E24">A9</font></i></a>, <a class="txt" href="glib_002.html#E1:33_1_2_1_1_1_2"><i><font color="Green" title="E31">A23</font></i></a>, <a class="txt" href="glib_002.html#E7:33_1_2"><i><font color="Green" title="E28">A13</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E4:33_1_2_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> 
 <b>by </b><i><a class="txt" href="glib_002.html#E2:33_1_2_1_1_1_2"><i><font color="Green" title="E32">A24</font></i></a>, <a class="ref" href="glib_001.html#T11" title="GLIB_001:th.11">GLIB_001:11</a></i>;<br/><a NAME="E5:33_1_2_1_1_1_2"/><b>then </b><i><font color="Green" title="E34">A26</font></i>: 
<font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">T</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33"><i><font color="Green" title="E18">A2</font></i></a>, <a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="glib_002.html#E2:33_1_2_1"><i><font color="Green" title="E30">A15</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:33_1_2_1_1_1_2"/><i><font color="Green" title="E35">A27</font></i>: 
<font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><a NAME="E7:33_1_2_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c14">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c12">T</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2_1_1_1_2"><i><font color="Green" title="E33">A25</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/><b>hence </b><a NAME="E8:33_1_2_1_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="glib_002.html#E3:33_1_2"><i><font color="Green" title="E26">A11</font></i></a>, <a class="txt" href="glib_002.html#E5:33_1_2_1_1_1_2"><i><font color="Green" title="E34">A26</font></i></a>, <a class="txt" href="glib_002.html#E6:33_1_2_1_1_1_2"><i><font color="Green" title="E35">A27</font></i></a>, <a class="ref" href="glib_001.html#T139" title="GLIB_001:th.139">GLIB_001:139</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:33_1_2_1"/>
contradiction
 ;<br/></div><b>end;</b></div><a NAME="E9:33_1_2"/><b>then </b>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_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">(<span class="default"><font color="Maroon" title="c2">v</font> <a href="glib_000.html#K29" title="GLIB_000:func.29">.edgesInOut()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_002.html#E1:33_1_2"><i><font color="Green" title="E25">A10</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E10:33_1_2"/><b>then </b>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_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">(<span class="default"><font color="Maroon" title="c1">G1</font> <a href="glib_000.html#K21" title="GLIB_000:func.21">.edgesInOut</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#D42" title="GLIB_000:def.42">GLIB_000:def 42</a></i>;<br/><a NAME="E11:33_1_2"/><b>then </b>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <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="c2">v</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="glib_000.html#T38" title="GLIB_000:th.38">GLIB_000:38</a></i>;<br/><b>hence </b><a NAME="E12:33_1_2"/>
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="ref" href="glib_000.html#T50" title="GLIB_000:th.50">GLIB_000:50</a></i>;<br/></div><b>end;</b></div><a NAME="E14:33_1"/><b>then </b>
<font color="Maroon" title="c12">T</font> <a href="glib_001.html#K14" title="GLIB_001:func.14">.edges()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c3">G2</font>
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c13">W'</font> = <font color="Maroon" title="c12">T</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G2</font> <b>by </b><i><a class="txt" href="glib_002.html#E11:33_1"><i><font color="Green" title="E23">A8</font></i></a>, <a class="ref" href="glib_001.html#T171" title="GLIB_001:th.171">GLIB_001:171</a></i>;<br/><a NAME="E16:33_1"/>
<font color="Maroon" title="c13">W'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v1'</font>,<font color="Maroon" title="c8">v2'</font>
 <b>by </b><i><a class="txt" href="glib_002.html#E9:33_1"><i><font color="Green" title="E22">A6</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>;<br/><b>hence </b><a NAME="E17:33_1"/>
 ex <font color="Olive" title="b1">W</font> being   <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c3">G2</font> st <font color="Olive" title="b1">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c7">v1'</font>,<font color="Maroon" title="c8">v2'</font>
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:33"/>
<font color="Maroon" title="c3">G2</font> is <a href="glib_002.html#V1" title="GLIB_002:attr.1">connected</a>
 <b>by </b><i><a class="ref" href="glib_002.html#D1" target="_self" title="GLIB_002:def.1">Def1</a></i>;<br/>


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