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<b>let </b><font color="Maroon">G</font> be  <a href="glib_000.html#V2">finite</a> <a href="glib_000.html#NM2">_Graph</a>;<br/><b>let </b><font color="Maroon">W</font> be   <a href="glib_001.html#NM4">Trail</a> of <font color="Maroon">G</font>;<br/>



<b>consider </b><font color="Maroon">f</font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E2:224"/><i><font color="Green">E39</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  &amp; ( 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> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  holds <br/><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Olive">x</font> <a href="mcart_1.html#K2">`2</a>  ) )
 <b>from </b><i><a class="ref" href="funct_1.html#S3">FUNCT_1:sch 3</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="224_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:224_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:224_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:224_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a></i>;<br/><b>consider </b><font color="Maroon">a</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:224_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><a NAME="E7:224_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/><a NAME="E8:224_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E9:224_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 ;<br/></div><b>end;</b></div>
<a NAME="E4:224"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font>
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="224_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">x2</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:224_2"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> )
 ;<br/><a NAME="E2:224_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x1</font> <a href="mcart_1.html#K2">`2</a>  &amp; <font color="Maroon">x2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font> <a href="mcart_1.html#K2">`2</a>  )
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a></i>;<br/><b>consider </b><font color="Maroon">a1</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b1</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:224_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a1</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><b>consider </b><font color="Maroon">a2</font> being    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:224_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">x2</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a2</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/><i><font color="Green">E49</font></i>: <font color="Maroon">b1</font> = 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x1</font>
<b>by </b><i><a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E50"><i><font color="Green">Lemma153</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/><a NAME="E8:224_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="glib_001.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E9:224_2"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><a NAME="E10:224_2"/>
( <font color="Maroon">a2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b1</font> )
 <b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E49"><i><font color="Green">Lemma152</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="ref" href="funct_1.html#T8">FUNCT_1:8</a></i>;<br/><b>hence </b><a NAME="E11:224_2"/>
<font color="Maroon">x1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x2</font>
 <b>by </b><i><a class="txt" href="glib_001.html#E51"><i><font color="Green">Lemma154</font></i></a>, <a class="txt" href="glib_001.html#E52"><i><font color="Green">Lemma155</font></i></a>, <a class="txt" href="glib_001.html#E53"><i><font color="Green">Lemma156</font></i></a>, <a class="txt" href="glib_001.html#E54"><i><font color="Green">Lemma157</font></i></a>, <a class="txt" href="glib_001.html#E55"><i><font color="Green">Lemma158</font></i></a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:224"/><b>then </b>
<font color="Maroon">f</font> is <a href="funct_1.html#V2">one-to-one</a>
 
<b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E7:224"/><b>then </b>
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="card_1.html#NR1" target="_self">&lt;=`</a>  <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="txt" href="glib_001.html#E47"><i><font color="Green">Lemma150</font></i></a>, <a class="txt" href="glib_001.html#E48"><i><font color="Green">Lemma151</font></i></a>, <a class="ref" href="card_1.html#T26">CARD_1:26</a></i>;<br/>
<a NAME="E8:224"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="card_2.html#T57">CARD_2:57</a></i>;<br/>
<b>hence </b><a NAME="E9:224"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12" target="_self">.edgeSeq()</a> </span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">G</font> <a href="glib_000.html#K18">.size()</a> 
 <b>by </b><i><a class="ref" href="glib_000.html#D27">GLIB_000:def 27</a></i>;<br/>


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