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<b>set </b><font color="Maroon">D</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:53_1_1_1"/>
not <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a>
 ;<br/><a NAME="E2:53_1_1_1"/><b>then </b>
<font color="Maroon">W</font> <a href="glib_001.html#K14">.edges()</a>  <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#T137">GLIB_001:137</a></i>;<br/><a NAME="E3:53_1_1_1"/><b>then </b>
 <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">.edgeSeq()</a> </span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="ref" href="glib_001.html#D17">GLIB_001:def 17</a></i>;<br/><b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E5:53_1_1_1"/><i><font color="Green">E37</font></i>: 
<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">.edgeSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:53_1_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">x</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">.edgeSeq()</a> </span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_001.html#K12">.edgeSeq()</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:53_1_1_1"/>
<font color="Maroon">x</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_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 <b>by </b><i>, </i>;<br/><a NAME="E9:53_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">x</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_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">D</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">W</font> <a href="glib_005.html#K8" target="_self">.flowSeq()</a> </span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="real_1.html#NM1">Real</a>) <b>-&gt; </b>  <a href="real_1.html#NM1">Real</a> = <font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">IT</font> being    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font><b> such that </b><br/><a NAME="E12:53_1_1_1"/><i><font color="Green">E82</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">D</font> holds  <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">IT</font>) <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">k</font>)
 <b>from </b><i><a class="ref" href="graph_5.html#S2">GRAPH_5:sch 2</a>();<br/></i><b>reconsider </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font> as   <a href="real_1.html#NM1">Real</a> ;<br/><b>take </b><font color="Maroon">IT</font> = <font color="Maroon">IT</font>;<br/><b>thus </b><a NAME="E14:53_1_1_1"/>
<font color="Maroon">IT</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_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 ;<br/><b>let </b><font color="Maroon">k</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E15:53_1_1_1"/>
<font color="Maroon">k</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_005.html#K8" target="_self">.flowSeq()</a> </span>)</span>
 ;<br/><b>hence </b><a NAME="E16:53_1_1_1"/>
<font color="Maroon">IT</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i/>;<br/></div>
<b>end;</b></div>

<b>thus </b><a NAME="E2:53_1_1"/>
( <font color="Maroon">W</font> is <a href="glib_001.html#V3">trivial</a> implies  ex <font color="Olive">b<sub>1</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 )
 ;<br/>


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