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<b>let </b><font color="Maroon" title="c1">G1</font> be  <a href="glib_000.html#V2" title="GLIB_000:attr.2">finite</a> <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</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">G2</font> being   <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">v1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b3">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">G2</font>  st <font color="Olive" title="b2">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">v2</font> holds <br/>( <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  &amp; <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Olive" title="b3">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  )</span><br/><b>let </b><font color="Maroon" title="c2">G2</font> be    <a href="glib_000.html#M2" title="GLIB_000:mode.2">Subgraph</a> of <font color="Maroon" title="c1">G1</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">v1</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font><br/> for <font color="Olive" title="b2">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">v2</font> holds <br/>( <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  &amp; <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Olive" title="b2">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  )</span><br/><b>let </b><font color="Maroon" title="c3">v1</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</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">v2</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">v2</font> holds <br/>( <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  &amp; <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Olive" title="b1">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  )</span><br/><b>let </b><font color="Maroon" title="c4">v2</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font> implies ( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  ) )</span><br/>







<b>assume </b><a NAME="E1:173"/><i><font color="Green" title="E70">A1</font></i>: 
<font color="Maroon" title="c3">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">v2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  )</span><br/>

<a NAME="E2:173"/><b>then </b>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K27" title="GLIB_000:func.27">.edgesIn()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="glib_000.html#T82" target="_self" title="GLIB_000:th.82">Th82</a></i>;<br/>
<b>hence </b><a NAME="E3:173"/><i><font color="Green" title="E71">A2</font></i>: 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K33" title="GLIB_000:func.33">.inDegree()</a> 
 <b>by </b><i><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  &amp; <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  )</span><br/>

<a NAME="E4:173"/><i><font color="Green" title="E72">A3</font></i>: 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="glib_000.html#K26" title="GLIB_000:func.26">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E1:173"><i><font color="Green" title="E70">A1</font></i></a>, <a class="ref" href="glib_000.html#T82" target="_self" title="GLIB_000:th.82">Th82</a></i>;<br/>
<a NAME="E5:173"/><b>then </b><i><font color="Green" title="E73">A4</font></i>: 
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="card_1.html#K5" title="CARD_1:func.5">card</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K28" title="GLIB_000:func.28">.edgesOut()</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<b>thus </b><a NAME="E6:173"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.outDegree()</a> 
 <b>by </b><i><a class="txt" href="#E4:173"><i><font color="Green" title="E72">A3</font></i></a>, <a class="ref" href="nat_1.html#T44" title="NAT_1:th.44">NAT_1:44</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a> </span><br/>

<b>thus </b><a NAME="E7:173"/>
<font color="Maroon" title="c4">v2</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c3">v1</font> <a href="glib_000.html#K36" title="GLIB_000:func.36">.degree()</a> 
 <b>by </b><i><a class="txt" href="#E3:173"><i><font color="Green" title="E71">A2</font></i></a>, <a class="txt" href="#E5:173"><i><font color="Green" title="E73">A4</font></i></a>, <a class="ref" href="xreal_1.html#T9" title="XREAL_1:th.9">XREAL_1:9</a></i>; <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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