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<span class="kw">let </span><font color="Maroon" title="c1">G1</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">E</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G2</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">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="b2">G2</font> st <br/>( <font color="Olive" title="b3">F</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">G1</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">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">G2</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">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">G2</font> st <br/>( <font color="Olive" title="b2">F</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">G1</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">G2</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">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">  ex <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="c3">G2</font> st <br/>( <font color="Olive" title="b1">F</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">G1</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )</span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">F</font> =  <a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c1">G1</font> as    <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="c3">G2</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#T12" onmouseover="rs('glib_010/T12')" onmouseout="rh()">GLIB_010:12</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c4">F</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="c4">F</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">G1</font> &amp; <font color="Maroon" title="c4">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:100"/>
<font color="Maroon" title="c4">F</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">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="c4">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a> </span><br/>

<a NAME="E3:100"/>
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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> &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G1</font> )
 
;<br/>
<a NAME="E4:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A1</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D11" onmouseover="rs('glib_010/D11')" onmouseout="rh()">GLIB_010:def 11</a></span>;<br/>
<a NAME="E5:100"/>
(  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">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="c3">G2</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c3">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T4" onmouseover="rs('glib_007/T4')" onmouseout="rh()">GLIB_007:4</a></span>;<br/>
<a NAME="E6:100"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A2</font></span>: 
<font color="Maroon" title="c4">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D12" onmouseover="rs('glib_010/D12')" onmouseout="rh()">GLIB_010:def 12</a></span>;<br/>
<a NAME="E7:100"/>
( <font color="Maroon" title="c4">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c4">F</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/>
<a NAME="E8:100"/><span class="kw">then </span>
<font color="Maroon" title="c4">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D13" onmouseover="rs('glib_010/D13')" onmouseout="rh()">GLIB_010:def 13</a></span>;<br/>
<span class="kw">hence </span><a NAME="E9:100"/>
<font color="Maroon" title="c4">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="#E4:100"><span class="lab"><font color="Green" title="E24">A1</font></span></a>, <a class="txt" href="#E6:100"><span class="lab"><font color="Green" title="E25">A2</font></span></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/>


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