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<b>let </b><font color="Maroon" title="c1">G</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_003.html#V7" title="GLIB_003:attr.7">real-weighted</a> <a href="glib_003.html#NM1" title="GLIB_003:NM.1">WGraph</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">EL</font> being  <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</font><br/> for <font color="Olive" title="b2">source</font>, <font color="Olive" title="b3">sink</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b4">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Olive" title="b1">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font> &amp; <font color="Olive" title="b2">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">V</font> &amp; not <font color="Olive" title="b3">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b4">V</font> holds <br/><font color="Olive" title="b1">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Olive" title="b2">source</font>,<font color="Olive" title="b3">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Olive" title="b4">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b4">V</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c2">EL</font> be   <a href="glib_005.html#NM1" title="GLIB_005:NM.1">FF:ELabeling</a> of <font color="Maroon" title="c1">G</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">source</font>, <font color="Olive" title="b2">sink</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b3">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> &amp; <font color="Olive" title="b1">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> &amp; not <font color="Olive" title="b2">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b3">V</font> holds <br/><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Olive" title="b1">source</font>,<font color="Olive" title="b2">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Olive" title="b3">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b3">V</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c3">source</font>, <font color="Maroon" title="c4">sink</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">V</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>  st <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> &amp; not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> holds <br/><font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Olive" title="b1">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Olive" title="b1">V</font></span>)</span></span>)</span></span>)</span></span><br/><b>let </b><font color="Maroon" title="c5">V</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> implies <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> )</span><br/>







<b>assume </b><a NAME="E1:43"/><i><font color="Green" title="E36">A1</font></i>: 
( <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#R1" title="GLIB_005:pred.1">has_valid_flow_from</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> &amp; <font color="Maroon" title="c3">source</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> &amp; not <font color="Maroon" title="c4">sink</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">V</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span><br/>

<b>set </b><font color="Maroon" title="c6">E1</font> = <font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">E2</font> = <font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c8">W1</font> = <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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c9">B1</font> =  <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span>;<br/>
<a NAME="E2:43"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E36">A1</font></i></a>, <a class="ref" href="glib_005.html#T11" target="_self" title="GLIB_005:th.11">Th11</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span> implies <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> )</span><br/><b>assume </b><a NAME="E1:43_1"/><i><font color="Green" title="E38">A3</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font></span><br/><a NAME="E2:43_1"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">EL</font> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>;<br/><b>hence </b><a NAME="E3:43_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="#E1:43"><i><font color="Green" title="E36">A1</font></i></a>, <a class="txt" href="#E1:43_1"><i><font color="Green" title="E38">A3</font></i></a>, <a class="ref" href="glib_005.html#D2" target="_self" title="GLIB_005:def.2">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:43"/><b>then </b><i><font color="Green" title="E38">A4</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_004.html#T5" title="GLIB_004:th.5">GLIB_004:5</a></i>;<br/>
<a NAME="E5:43"/><i><font color="Green" title="E39">A5</font></i>: 
 <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="uproots.html#T13" title="UPROOTS:th.13">UPROOTS:13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="43_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font> implies <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> )</span><br/><b>assume </b><a NAME="E1:43_2"/><i><font color="Green" title="E40">A6</font></i>: 
<font color="Maroon" title="c10">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font></span><br/><a NAME="E2:43_2"/>
 <a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span> <a href="pboole.html#R6" title="PBOOLE:pred.6">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span> <a href="pre_circ.html#K1" title="PRE_CIRC:func.1">--&gt;</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>by </b><i><a class="ref" href="polynom1.html#D15" title="POLYNOM1:def.15">POLYNOM1:def 15</a></i>;<br/><b>hence </b><a NAME="E3:43_2"/>
<span class="p1">(<span class="default"><a href="polynom1.html#K14" title="POLYNOM1:func.14">EmptyBag</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c10">e</font>
 <b>by </b><i><a class="txt" href="#E1:43_2"><i><font color="Green" title="E40">A6</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E7:43"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E5:43"><i><font color="Green" title="E39">A5</font></i></a>, <a class="ref" href="glib_004.html#T5" title="GLIB_004:th.5">GLIB_004:5</a></i>;<br/>
<a NAME="E8:43"/><b>then </b><i><font color="Green" title="E40">A8</font></i>: 
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<b>by </b><i><a class="ref" href="xreal_1.html#T15" title="XREAL_1:th.15">XREAL_1:15</a></i>;<br/>
<a NAME="E9:43"/>
<span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><span class="p3">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <span class="p1">(<span class="default"><a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">EL</font> <a href="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p3">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span>,<font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E4:43"><i><font color="Green" title="E38">A4</font></i></a>, <a class="ref" href="xreal_1.html#T11" title="XREAL_1:th.11">XREAL_1:11</a></i>;<br/>
<b>hence </b><a NAME="E10:43"/>
<font color="Maroon" title="c2">EL</font> <a href="glib_005.html#K1" title="GLIB_005:func.1">.flow</a> <font color="Maroon" title="c3">source</font>,<font color="Maroon" title="c4">sink</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="uproots.html#K3" title="UPROOTS:func.3">Sum</a> <span class="p1">(<span class="default"><span class="p2">(<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="pre_circ.html#K2" title="PRE_CIRC:func.2">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_000.html#K24" title="GLIB_000:func.24">.edgesDBetween</a> <font color="Maroon" title="c5">V</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font></span>)</span> <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <font color="Maroon" title="c5">V</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:43"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E8:43"><i><font color="Green" title="E40">A8</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
