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

<span class="kw">let </span><font color="Maroon" title="c1">G3</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">V</font>, <font color="Olive" title="b2">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G4</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G3</font>,<font color="Olive" title="b2">E</font><br/>  for <font color="Olive" title="b4">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G3</font>,<font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b5">G2</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Olive" title="b3">G4</font>,<font color="Olive" title="b1">V</font> holds  <font color="Olive" title="b5">G2</font> is <font color="Olive" title="b4">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font>, <font color="Maroon" title="c3">E</font> be    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</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">G4</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c3">E</font><br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b3">G2</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Olive" title="b1">G4</font>,<font color="Maroon" title="c2">V</font> holds  <font color="Olive" title="b3">G2</font> is <font color="Olive" title="b2">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">G4</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c3">E</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">G1</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c2">V</font><br/>  for <font color="Olive" title="b2">G2</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c4">G4</font>,<font color="Maroon" title="c2">V</font> holds  <font color="Olive" title="b2">G2</font> is <font color="Olive" title="b1">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c5">G1</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c2">V</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">G2</font> being    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c4">G4</font>,<font color="Maroon" title="c2">V</font> holds  <font color="Olive" title="b1">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c6">G2</font> be    <a href="glib_012.html#M1" title="GLIB_012:mode.1">addLoops</a> of <font color="Maroon" title="c4">G4</font>,<font color="Maroon" title="c2">V</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="c6">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/>









<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:58_1"/>
( <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</font> or  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_1_1"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</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="c6">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c7">F0</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c4">G4</font><span class="kw"> such that </span><br/><a NAME="E3:58_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
( <font color="Maroon" title="c7">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c1">G3</font> &amp; <font color="Maroon" title="c7">F0</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#T77" onmouseover="rs('glibpre0/T77')" onmouseout="rh()">GLIBPRE0:77</a></span>;<br/><a NAME="E4:58_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">V</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="funct_1.html#T52" onmouseover="rs('funct_1/T52')" onmouseout="rh()">FUNCT_1:52</a></span>;<br/><span class="lab"><font color="Green" title="E32">A4</font></span>:  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">V</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T61" onmouseover="rs('relat_1/T61')" onmouseout="rh()">RELAT_1:61</a></span>
<br/>.= 
<font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:58_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="xboole_1.html#T28" onmouseover="rs('xboole_1/T28')" onmouseout="rh()">XBOOLE_1:28</a></span>
;<br/><span class="lab"><font color="Green" title="E33">A5</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c2">V</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c1">G3</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="ref" href="relat_1.html#T115" onmouseover="rs('relat_1/T115')" onmouseout="rh()">RELAT_1:115</a></span>
<br/>.= 
<font color="Maroon" title="c2">V</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="funct_1.html#T92" onmouseover="rs('funct_1/T92')" onmouseout="rh()">FUNCT_1:92</a></span>
;<br/><span class="kw">consider </span><font color="Maroon" title="c8">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c6">G2</font><span class="kw"> such that </span><br/><a NAME="E8:58_1_1"/>
( <font color="Maroon" title="c8">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c8">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  )
 <span class="kw">and </span><br/><a NAME="E9:58_1_1"/>
(  not <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E10:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  )
 <span class="kw">and </span><br/><a NAME="E11:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  )
 <span class="kw">and </span><br/><a NAME="E12:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  )
 <span class="kw">and </span><br/><a NAME="E13:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  )
 <span class="kw">and </span><br/><a NAME="E14:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c8">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="E15:58_1_1"/><span class="lab"><font color="Green" title="E34">A6</font></span>: 
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )
 <span class="kw">and </span><br/><a NAME="E16:58_1_1"/>
( <font color="Maroon" title="c7">F0</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a>  implies <font color="Maroon" title="c8">F</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_1_1"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E4:58_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="txt" href="#E5:58_1_1"><span class="lab"><font color="Green" title="E32">A4</font></span></a>, <a class="txt" href="#E6:58_1_1"><span class="lab"><font color="Green" title="E33">A5</font></span></a>, <a class="ref" href="glib_012.html#T29" target="_self" onmouseover="rs('glib_012/T29')" onmouseout="rh()">Th29</a></span>;<br/><span class="kw">thus </span><a NAME="E17:58_1_1"/>
<font color="Maroon" title="c6">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_1"><span class="lab"><font color="Green" title="E30">A2</font></span></a>, <a class="txt" href="#E15:58_1_1"><span class="lab"><font color="Green" title="E34">A6</font></span></a>, <a class="ref" href="glib_010.html#D23" onmouseover="rs('glib_010/D23')" onmouseout="rh()">GLIB_010:def 23</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_1_2"/>
 not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</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="c6">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><div class="add"><a NAME="E2:58_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c5">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G3</font> &amp;  not <font color="Maroon" title="c2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a>, <a class="ref" href="glib_007.html#T4" onmouseover="rs('glib_007/T4')" onmouseout="rh()">GLIB_007:4</a></span>;<br/><a NAME="E3:58_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E29">A7</font></span>: 
( <font color="Maroon" title="c5">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G3</font> &amp; <font color="Maroon" title="c6">G2</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c4">G4</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#D5" target="_self" onmouseover="rs('glib_012/D5')" onmouseout="rh()">Def5</a></span>;<br/><a NAME="E4:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">G2</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G3</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T2" onmouseover="rs('glib_007/T2')" onmouseout="rh()">GLIB_007:2</a></span>;<br/><a NAME="E5:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c1">G3</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T3" onmouseover="rs('glib_007/T3')" onmouseout="rh()">GLIB_007:3</a></span>;<br/><a NAME="E6:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G1</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c6">G2</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_1_2"><span class="lab"><font color="Green" title="E29">A7</font></span></a>, <a class="ref" href="glib_007.html#T2" onmouseover="rs('glib_007/T2')" onmouseout="rh()">GLIB_007:2</a></span>;<br/><a NAME="E7:58_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c6">G2</font> is    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c5">G1</font>,<font color="Maroon" title="c3">E</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T3" onmouseover="rs('glib_007/T3')" onmouseout="rh()">GLIB_007:3</a></span>;<br/><span class="kw">hence </span><a NAME="E8:58_1_2"/>
<font color="Maroon" title="c6">G2</font> is <font color="Maroon" title="c5">G1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T78" onmouseover="rs('glibpre0/T78')" onmouseout="rh()">GLIBPRE0:78</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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