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

<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font><br/>  for <font color="Olive" title="b2">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>  st <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  holds <br/><font color="Olive" title="b2">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b2">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F0</font> be    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>  st <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  holds <br/><font color="Olive" title="b1">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">v</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:114"/><span class="lab"><font color="Green" title="E32">A1</font></span>: 
<font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c4">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a> </span><br/>

<a NAME="E2:114"/><span class="kw">then </span>
<font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">E</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#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:114"/><span class="lab"><font color="Green" title="E33">A2</font></span>: 
 for <font color="Olive" title="b1">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">E</font>  ex <font color="Olive" title="b2">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T87" target="_self" onmouseover="rs('glibpre0/T87')" onmouseout="rh()">Th93</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">G3</font> =  the    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">E</font>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>, the    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">E</font><span class="kw"> such that </span><br/><a NAME="E6:114"/><span class="lab"><font color="Green" title="E34">A3</font></span>: 
( <font color="Maroon" title="c7">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c7">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  )
 <span class="kw">and </span><br/><a NAME="E7:114"/><span class="lab"><font color="Green" title="E35">A4</font></span>: 
( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c7">F</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  )
 <span class="kw">and </span><br/><a NAME="E8:114"/>
( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  implies <font color="Maroon" title="c7">F</font> is  <a href="glib_010.html#V16" title="GLIB_010:attr.16">strong_SG-embedding</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c7">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:114"><span class="lab"><font color="Green" title="E33">A2</font></span></a></span>;<br/>
<a NAME="E9:114"/><span class="lab"><font color="Green" title="E36">A5</font></span>: 
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E7:114"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="txt" href="glibpre0.html#E97"><span class="lab"><font color="Green" title="E31">Lm5</font></span></a></span>;<br/>
<a NAME="E10:114"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:114"><span class="lab"><font color="Green" title="E32">A1</font></span></a>, <a class="txt" href="#E7:114"><span class="lab"><font color="Green" title="E35">A4</font></span></a>, <a class="ref" href="glib_010.html#D11" onmouseover="rs('glib_010/D11')" onmouseout="rh()">GLIB_010:def 11</a></span>;<br/>
<a NAME="E11:114"/><span class="kw">then </span><span class="lab"><font color="Green" title="E37">A6</font></span>: 
( <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Maroon" title="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E34">A3</font></span></a></span>;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:114"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:114"><span class="lab"><font color="Green" title="E34">A3</font></span></a>, <a class="txt" href="#E11:114"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E13:114"/>
<font color="Maroon" title="c4">v</font> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c4">v</font></span>)</span> <a href="glib_000.html#K34" title="GLIB_000:func.34">.degree()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:114"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="ref" href="glibpre0.html#T54" target="_self" onmouseover="rs('glibpre0/T54')" onmouseout="rh()">Th60</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
