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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2" target="_self">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">v1</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>, <font color="Maroon">v2</font> be   <a href="glib_000.html#NM10">Vertex</a> of <font color="Maroon">G</font>;<br/>





<b>assume </b><a NAME="E1:161"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">e</font> <a href="glib_000.html#R1" target="_self">Joins</a> <font color="Maroon">v1</font>,<font color="Maroon">v2</font>,<font color="Maroon">G</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:161_1_1"/>
( ( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  ) or ( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  ) )
 </span><b>by </b><i>, </i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_1_1"><b>suppose </b></a><a NAME="E1:161_1_1_1"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  )
 ;<br/><div class="add"><a NAME="E2:161_1_1_1"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:161_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:161_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E5:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E6:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> 
 ;<br/><a NAME="E7:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E8:161_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K36" target="_self">.degree()</a> 
 ;<br/><a NAME="E9:161_1_1_1"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:161_1_1_1"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E11:161_1_1_1"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E12:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E13:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> 
 ;<br/><a NAME="E14:161_1_1_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E15:161_1_1_1"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K36" target="_self">.degree()</a> 
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_1_2"><b>suppose </b></a><a NAME="E1:161_1_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a>  &amp; <font color="Maroon">e</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a>  )
 ;<br/><div class="add"><a NAME="E2:161_1_1_2"/><b>then </b>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:161_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E4:161_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E5:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K28" target="_self">.edgesOut()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E6:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> 
 ;<br/><a NAME="E7:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v1</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E8:161_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K36" target="_self">.degree()</a> 
 ;<br/><a NAME="E9:161_1_1_2"/>
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> 
 <b>by </b><i>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E10:161_1_1_2"/><b>then </b>
<span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><a NAME="E11:161_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">e</font></span><a href="tarski.html#K1">}</a></span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T80">CARD_1:80</a></i>;<br/><a NAME="E12:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K27" target="_self">.edgesIn()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="card_1.html#T79">CARD_1:79</a></i>;<br/><a NAME="E13:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> 
 ;<br/><a NAME="E14:161_1_1_2"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K34" target="_self">.outDegree()</a> </span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">v2</font> <a href="glib_000.html#K33" target="_self">.inDegree()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><b>hence </b><a NAME="E15:161_1_1_2"/>
1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K36" target="_self">.degree()</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:161"/>
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v1</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">v2</font> <a href="glib_000.html#K36" target="_self">.degree()</a>  )
 ;<br/>


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