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

<span class="kw">let </span><font color="Maroon" title="c1">G</font> be   <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">G</font> is  <a href="glib_005.html#V1" title="GLIB_005:attr.1">natural-weighted</a>  implies <font color="Maroon" title="c1">G</font> is  <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:6_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_005.html#V1" title="GLIB_005:attr.1">natural-weighted</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">G</font> is  <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a> </span><br/>

<a NAME="E2:6_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A1</font></span>: 
 <a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font> is  <a href="valued_0.html#V6" title="VALUED_0:attr.6">natural-valued</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_005.html#D1" target="_self" title="GLIB_005:def.1">Def1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="6_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c2">y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> implies <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a>  )</span><br/><span class="kw">assume </span><a NAME="E1:6_1_1_1"/><span class="lab"><font color="Green" title="E7">A2</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> </span><br/><a NAME="E2:6_1_1_1"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:6_1_1"><span class="lab"><font color="Green" title="E6">A1</font></span></a>, <a class="ref" href="valued_0.html#D6" title="VALUED_0:def.6">VALUED_0:def 6</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">y9</font> = <font color="Maroon" title="c2">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1_1_1"><span class="lab"><font color="Green" title="E7">A2</font></span></a></span>;<br/><a NAME="E4:6_1_1_1"/>
<font color="Maroon" title="c3">y9</font> is   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>
 ;<br/><span class="kw">hence </span><a NAME="E5:6_1_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="graph_5.html#D12" title="GRAPH_5:def.12">GRAPH_5:def 12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:6_1_1"/><span class="kw">then </span>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="glib_003.html#K5" title="GLIB_003:func.5">the_Weight_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="graph_5.html#K8" title="GRAPH_5:func.8">Real&gt;=0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:6_1_1"/>
<font color="Maroon" title="c1">G</font> is  <a href="glib_003.html#V8" title="GLIB_003:attr.8">nonnegative-weighted</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_003.html#D14" title="GLIB_003:def.14">GLIB_003:def 14</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
