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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">s</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G</font> holds   <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">s</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Olive" title="b1">s</font></span><br/><b>let </b><font color="Maroon" title="c2">src</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</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">  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span><br/>



<b>set </b><font color="Maroon" title="c3">Gn</font> =  <a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font>;<br/>
<b>set </b><font color="Maroon" title="c4">RFS</font> = <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>;<br/>
<b>set </b><font color="Maroon" title="c5">DCS</font> =  <a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font>;<br/>
<b>set </b><font color="Maroon" title="c6">n</font> = <span class="p1">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> ;<br/>
<i><font color="Green" title="E37">A1</font></i>:  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span></span>)</span> = 
 <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_004.html#K10" title="GLIB_004:func.10">DIJK:CompSeq</a> <font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_000.html#K40" title="GLIB_000:func.40">.Lifespan()</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T21" target="_self" title="GLIB_004:th.21">Th20</a></i>
<br/>.= 
 <a href="rfunct_3.html#K1" title="RFUNCT_3:func.1">min</a> <span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></span>)</span>,<span class="p1">(<span class="default"><a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p2">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="glib_004.html#T25" target="_self" title="GLIB_004:th.25">Th24</a></i>
<br/>.= 
 <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font></span>)</span>

;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_2"/><i><font color="Green" title="E38">A2</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:51_2"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="ref" href="glib_004.html#T19" target="_self" title="GLIB_004:th.19">Th18</a></i>;<br/><a NAME="E3:51_2"/><b>then </b>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 <b>by </b><i><a class="txt" href="#E1:51_2"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></i>;<br/><b>hence </b><a NAME="E4:51_2"/>
contradiction
 <b>by </b><i><a class="txt" href="#E1:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="trees_1.html#T24" title="TREES_1:th.24">TREES_1:24</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>
<b>hence </b><a NAME="E3:51"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_004.html#K11" title="GLIB_004:func.11">DIJK:SSSP</a> <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">src</font></span>)</span> <a href="glib_004.html#K5" title="GLIB_004:func.5">`1</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">G</font> <a href="glib_002.html#K2" title="GLIB_002:func.2">.reachableDFrom</a> <font color="Maroon" title="c2">src</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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