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<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">G4</font> being  <font color="Maroon" title="c1">G3</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><br/>  for <font color="Olive" title="b2">G1</font> being   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Maroon" title="c1">G3</font><br/>  for <font color="Olive" title="b3">G2</font> being   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Olive" title="b1">G4</font> holds  <font color="Olive" title="b3">G2</font> is <font color="Olive" title="b2">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">G4</font> be  <font color="Maroon" title="c1">G3</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a>  <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">G1</font> being   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Maroon" title="c1">G3</font><br/>  for <font color="Olive" title="b2">G2</font> being   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Maroon" title="c2">G4</font> holds  <font color="Olive" title="b2">G2</font> is <font color="Olive" title="b1">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">G1</font> be   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <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">  for <font color="Olive" title="b1">G2</font> being   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Maroon" title="c2">G4</font> holds  <font color="Olive" title="b1">G2</font> is <font color="Maroon" title="c3">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c4">G2</font> be   <a href="glib_012.html#NM2" title="GLIB_012:NM.2">addLoops</a> of <font color="Maroon" title="c2">G4</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">G2</font> is <font color="Maroon" title="c3">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> </span><br/>







<span class="kw">consider </span><font color="Maroon" title="c5">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="c2">G4</font><span class="kw"> such that </span><br/><a NAME="E2:56"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c5">F0</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="ref" href="glib_010.html#D24" onmouseover="rs('glib_010/D24')" onmouseout="rh()">GLIB_010:def 24</a></span>;<br/>
<a NAME="E3:56"/><span class="lab"><font color="Green" title="E30">A2</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="c5">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> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</font></span>)</span></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">G3</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E29">A1</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="E4:56"/>
 <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="c5">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> <span class="p2">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G3</font></span>)</span></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="c2">G4</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="ref" href="glib_010.html#D12" onmouseover="rs('glib_010/D12')" onmouseout="rh()">GLIB_010:def 12</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c3">G1</font>,<font color="Maroon" title="c4">G2</font><span class="kw"> such that </span><br/><a NAME="E6:56"/>
( <font color="Maroon" title="c6">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="c5">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="c6">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="c5">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="c5">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  )
 <span class="kw">and </span><br/><a NAME="E7:56"/>
(  not <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  )
 <span class="kw">and </span><br/><a NAME="E8:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  )
 <span class="kw">and </span><br/><a NAME="E9:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  )
 <span class="kw">and </span><br/><a NAME="E10:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">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="E11:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  )
 <span class="kw">and </span><br/><a NAME="E12:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c6">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="E13:56"/>
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c6">F</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )
 <span class="kw">and </span><br/><a NAME="E14:56"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
( <font color="Maroon" title="c5">F0</font> is  <a href="glib_010.html#V18" title="GLIB_010:attr.18">Disomorphism</a>  implies <font color="Maroon" title="c6">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="#E2:56"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E3:56"><span class="lab"><font color="Green" title="E30">A2</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="E15:56"/>
<font color="Maroon" title="c4">G2</font> is <font color="Maroon" title="c3">G1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:56"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E14:56"><span class="lab"><font color="Green" title="E31">A3</font></span></a>, <a class="ref" href="glib_010.html#D24" onmouseover="rs('glib_010/D24')" onmouseout="rh()">GLIB_010:def 24</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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