<?xml version="1.0"?>
<div class="add">

<b>consider </b><font color="Maroon">G1</font> being  <a href="glib_000.html#V2">finite</a> non <a href="glib_000.html#V4">trivial</a> <a href="glib_002.html#V3">Tree-like</a> <a href="glib_000.html#NM2">_Graph</a>;<br/>
<b>set </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0;<br/>
<a NAME="E2:89_2_1"/><i><font color="Green">E52</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font> &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:89_2_1_1"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:89_2_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/></div><b>end;</b></div>
<a NAME="E4:89_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E5:89_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">W</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font></span>)</span> <a href="pboole.html#K9">--&gt;</a> 0 as   <a href="funct_2.html#NM1">Function</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <font color="Maroon">G1</font>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<b>set </b><font color="Maroon">G2</font> = <font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font>;<br/>
<b>reconsider </b><font color="Maroon">EL</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="glib_000.html#K7">the_Edges_of</a> <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/>
<b>set </b><font color="Maroon">G3</font> = <span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font>;<br/>
<b>reconsider </b><font color="Maroon">VL</font> =  <a href="xboole_0.html#K1">{}</a>  as   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="glib_000.html#K6">the_Vertices_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span>, <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="partfun1.html#T56">PARTFUN1:56</a></i>;<br/>
<b>set </b><font color="Maroon">G4</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font>;<br/>
<b>take </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font>
;<br/>

<b>thus </b><a NAME="E9:89_2_1"/>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_000.html#V2">finite</a> &amp; not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_000.html#V4">trivial</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_002.html#V3">Tree-like</a> )
 ;<br/>

<i><font color="Green">E55</font></i>:  <a href="glib_003.html#K5" target="_self">the_Weight_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<font color="Maroon">W</font>
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
;<br/>
<a NAME="E11:89_2_1"/><b>then </b><i><font color="Green">E56</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_003.html#V7" target="_self">real-weighted</a>
 
<b>by </b><i/>;<br/>
 <a href="glib_003.html#K6" target="_self">the_ELabel_of</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="funct_1.html#K1">.</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> 
<b>by </b><i><a class="ref" href="glib_000.html#T12">GLIB_000:12</a></i>
<br/>.= 
<font color="Maroon">EL</font>
<b>by </b><i><a class="ref" href="glib_000.html#T11">GLIB_000:11</a></i>
;<br/>
<a NAME="E13:89_2_1"/><b>then </b><i><font color="Green">E57</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_003.html#V9" target="_self">real-elabeled</a>
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="89_2_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:89_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span>
 ;<br/><a NAME="E2:89_2_1_4"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a> 0
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E3:89_2_1_4"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 <b>by </b><i><a class="ref" href="graph_5.html#D12">GRAPH_5:def 12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E15:89_2_1"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default">0</span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 
<b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<a NAME="E16:89_2_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">W</font> <a href="tarski.html#R1">c=</a>  <a href="graph_5.html#K8">Real&gt;=0</a> 
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>hence </b><a NAME="E17:89_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_003.html#V8" target="_self">nonnegative-weighted</a>
 <b>by </b><i>, </i>;<br/>

<b>thus </b><a NAME="E18:89_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">G1</font> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K2" target="_self">WeightSelector</a> ,<font color="Maroon">W</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K3" target="_self">ELabelSelector</a> ,<font color="Maroon">EL</font></span>)</span> <a href="glib_000.html#K13">.set</a> <a href="glib_003.html#K4" target="_self">VLabelSelector</a> ,<font color="Maroon">VL</font> is <a href="glib_003.html#V11" target="_self">real-WEV</a>
 <b>by </b><i>, , </i>;<br/>


</div>
