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<b>let </b><font color="Maroon" title="c1">X</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">g</font> be    <a href="sgraph1.html#M1" title="SGRAPH1:mode.1">SimpleGraph</a> of <font color="Maroon" title="c1">X</font>;<br/><b>let </b><font color="Maroon" title="c3">v</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>





<b>assume </b><a NAME="E1:39"/><i><font color="Green" title="E23">A1</font></i>: 
<font color="Maroon" title="c3">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">VV</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> as  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  <b>by </b><i><a class="ref" href="sgraph1.html#T27" target="_self" title="SGRAPH1:th.27">Th27</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c4">VV</font>
;<br/>

<b>consider </b><font color="Maroon" title="c5">ww</font> being  <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:39"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c5">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">w</font> where <font color="Olive" title="b1">w</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font> : ( <font color="Olive" title="b1">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Olive" title="b1">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c2">g</font> ) </span>}</span>  &amp;  <a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <font color="Maroon" title="c5">ww</font> )
 <b>by </b><i><a class="ref" href="sgraph1.html#T39" target="_self" title="SGRAPH1:th.39">Th39</a></i>;<br/>
<a NAME="E5:39"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c5">ww</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c4">VV</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:39_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">ww</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c7">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:39_1"/><i><font color="Green" title="E26">A4</font></i>: 
( <font color="Maroon" title="c6">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">w</font> &amp; <font color="Maroon" title="c7">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c7">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:39"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/>
<b>thus </b><a NAME="E4:39_1"/>
<font color="Maroon" title="c6">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:39_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>


</div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="39_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:39_2"/>
<font color="Maroon" title="c5">ww</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">VV</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c6">w</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">X</font><b> such that </b><br/><a NAME="E3:39_2"/><i><font color="Green" title="E26">A5</font></i>: 
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">w</font> &amp; <font color="Maroon" title="c6">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c6">w</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="sgraph1.html#U1" title="SGRAPH1:sel.1">SEdges</a> of <font color="Maroon" title="c2">g</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E1:39"><i><font color="Green" title="E23">A1</font></i></a>, <a class="txt" href="sgraph1.html#E4:39"><i><font color="Green" title="E24">A2</font></i></a></i>;<br/><a NAME="E4:39_2"/>
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c3">v</font>,<font color="Maroon" title="c3">v</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <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="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="ref" href="enumset1.html#T69" title="ENUMSET1:th.69">ENUMSET1:69</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E6:39_2"/><i><font color="Green" title="E27">A6</font></i>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">VV</font> &amp; <font color="Maroon" title="c7">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">y</font> &amp; <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c3">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E3:39_2"><i><font color="Green" title="E26">A5</font></i></a>, <a class="ref" href="sgraph1.html#T10" target="_self" title="SGRAPH1:th.10">Th10</a></i>;<br/><a NAME="E7:39_2"/>
( <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> &amp; <font color="Maroon" title="c3">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">y</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E6:39_2"><i><font color="Green" title="E27">A6</font></i></a>, <a class="ref" href="zfmisc_1.html#T8" title="ZFMISC_1:th.8">ZFMISC_1:8</a></i>;<br/><b>hence </b><a NAME="E8:39_2"/>
<font color="Maroon" title="c5">ww</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">VV</font>
 <b>by </b><i><a class="txt" href="sgraph1.html#E6:39_2"><i><font color="Green" title="E27">A6</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E7:39"/><b>then </b>
<font color="Maroon" title="c5">ww</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c4">VV</font>
 
<b>by </b><i><a class="txt" href="sgraph1.html#E5:39"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/>
<b>hence </b><a NAME="E8:39"/>
( <font color="Maroon" title="c4">VV</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">g</font> &amp;  <a href="sgraph1.html#K4" title="SGRAPH1:func.4">degree</a> <font color="Maroon" title="c2">g</font>,<font color="Maroon" title="c3">v</font> <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> <font color="Maroon" title="c4">VV</font> )
 <b>by </b><i><a class="txt" href="sgraph1.html#E4:39"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="card_2.html#T67" title="CARD_2:th.67">CARD_2:67</a></i>;<br/>


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