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<span class="kw">let </span><font color="Maroon" title="c1">n</font> be   non  <a href="ordinal1.html#V8" title="ORDINAL1:attr.8">zero</a>  <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</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">C1</font>, <font color="Olive" title="b2">C2</font> being  <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>   <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a> holds  <font color="Olive" title="b2">C2</font> is <font color="Olive" title="b1">C1</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">C1</font>, <font color="Maroon" title="c3">C2</font> be  <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a>   <a href="glpacy00.html#V4" title="GLPACY00:attr.4">Cycle-like</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"> <font color="Maroon" title="c3">C2</font> is <font color="Maroon" title="c2">C1</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="108_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="E1:108_1"/>
( <font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1"><span class="kw">suppose </span></a><a NAME="E1:108_1_1"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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">C2</font> is <font color="Maroon" title="c2">C1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><div class="add"><a NAME="E2:108_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A1</font></span>: 
( <font color="Maroon" title="c2">C1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp; <font color="Maroon" title="c3">C2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 ;<br/><a NAME="E3:108_1_1"/>
( <font color="Maroon" title="c2">C1</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> &amp; <font color="Maroon" title="c3">C2</font> <a href="glib_013.html#K2" title="GLIB_013:func.2">.size()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">n</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D4" onmouseover="rs('glib_013/D4')" onmouseout="rh()">GLIB_013:def 4</a></span>;<br/><a NAME="E4:108_1_1"/><span class="kw">then </span>
 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="c2">C1</font>,<font color="Maroon" title="c3">C2</font> st <font color="Olive" title="b1">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="#E2:108_1_1"><span class="lab"><font color="Green" title="E52">A1</font></span></a>, <a class="ref" href="glib_010.html#T177" onmouseover="rs('glib_010/T177')" onmouseout="rh()">GLIB_010:177</a></span>;<br/><a NAME="E5:108_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c3">C2</font> is <font color="Maroon" title="c2">C1</font> <a href="glib_010.html#V20" title="GLIB_010:attr.20">-Disomorphic</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/><span class="kw">hence </span><a NAME="E6:108_1_1"/>
<font color="Maroon" title="c3">C2</font> is <font color="Maroon" title="c2">C1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</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"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_2"><span class="kw">suppose </span></a><a NAME="E1:108_1_2"/>
<font color="Maroon" title="c1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1
 ; <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">C2</font> is <font color="Maroon" title="c2">C1</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/><div class="add"><a NAME="E2:108_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c2">C1</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 &amp; <font color="Maroon" title="c3">C2</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 1 )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>;<br/><a NAME="E3:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E52">A2</font></span>: 
(  not <font color="Maroon" title="c2">C1</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  &amp;  not <font color="Maroon" title="c3">C2</font> is  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T26" onmouseover="rs('glib_000/T26')" onmouseout="rh()">GLIB_000:26</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c4">e1</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c5">e2</font> =  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c6">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c7">w1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>;<br/><span class="kw">set </span><font color="Maroon" title="c8">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>;<br/><span class="kw">set </span><font color="Maroon" title="c9">w2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>;<br/><a NAME="E4:108_1_2"/><span class="lab"><font color="Green" title="E53">A3</font></span>: 
(  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c2">C1</font> &amp;  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>,<font color="Maroon" title="c3">C2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">P1</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E6:108_1_2"/><span class="lab"><font color="Green" title="E54">A4</font></span>: 
(  not  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c10">P1</font> &amp; <font color="Maroon" title="c2">C1</font> is    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c10">P1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>, the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font> )
 <span class="kw">and </span><br/><a NAME="E7:108_1_2"/><span class="lab"><font color="Green" title="E55">A5</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c10">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:108_1_2"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="glpacy00.html#T37" target="_self" onmouseover="rs('glpacy00/T37')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font>, <font color="Maroon" title="c12">w1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c2">C1</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c10">P1</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="glpacy00.html#E64"><span class="lab"><font color="Green" title="E50">Lm8</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c13">P2</font> being   <a href="glib_000.html#V2" title="GLIB_000:attr.2">_finite</a>   non  <a href="glib_000.html#V4" title="GLIB_000:attr.4">_trivial</a>   <a href="glpacy00.html#V1" title="GLPACY00:attr.1">Path-like</a>  <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a><span class="kw"> such that </span><br/><a NAME="E10:108_1_2"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
(  not  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c13">P2</font> &amp; <font color="Maroon" title="c3">C2</font> is    <a href="glib_006.html#M3" title="GLIB_006:mode.3">addEdge</a> of <font color="Maroon" title="c13">P2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>, the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font> )
 <span class="kw">and </span><br/><a NAME="E11:108_1_2"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c13">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font></span>)</span></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:108_1_2"><span class="lab"><font color="Green" title="E52">A2</font></span></a>, <a class="txt" href="#E4:108_1_2"><span class="lab"><font color="Green" title="E53">A3</font></span></a>, <a class="ref" href="glpacy00.html#T37" target="_self" onmouseover="rs('glpacy00/T37')" onmouseout="rh()">Th45</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font>, <font color="Maroon" title="c15">w2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c3">C2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c13">P2</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="glpacy00.html#E64"><span class="lab"><font color="Green" title="E50">Lm8</font></span></a></span>;<br/><font color="Maroon" title="c10">P1</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  = 
<font color="Maroon" title="c2">C1</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:108_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="glpacy00.html#E64"><span class="lab"><font color="Green" title="E50">Lm8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>
;<br/><a NAME="E14:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c10">P1</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>;<br/><font color="Maroon" title="c13">P2</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a>  = 
<font color="Maroon" title="c3">C2</font> <a href="glib_013.html#K1" title="GLIB_013:func.1">.order()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="glpacy00.html#E64"><span class="lab"><font color="Green" title="E50">Lm8</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c1">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>
;<br/><a NAME="E16:108_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">P2</font> is <font color="Maroon" title="c1">n</font> <a href="glib_013.html#V3" title="GLIB_013:attr.3">-vertex</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_013.html#D3" onmouseover="rs('glib_013/D3')" onmouseout="rh()">GLIB_013:def 3</a></span>;<br/><a NAME="E17:108_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c13">P2</font> is <font color="Maroon" title="c10">P1</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="#E14:108_1_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="glpacy00.html#T24" target="_self" onmouseover="rs('glpacy00/T24')" onmouseout="rh()">Th32</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c16">F0</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c10">P1</font>,<font color="Maroon" title="c13">P2</font><span class="kw"> such that </span><br/><a NAME="E19:108_1_2"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
<font color="Maroon" title="c16">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="glib_010.html#D23" onmouseover="rs('glib_010/D23')" onmouseout="rh()">GLIB_010:def 23</a></span>;<br/><a NAME="E20:108_1_2"/><span class="lab"><font color="Green" title="E60">A10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">F0</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="c10">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E59">A9</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="E21:108_1_2"/>
<font color="Maroon" title="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_1_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span>1 = 
<font color="Maroon" title="c11">v1</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font></span>)</span> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="glibpre0.html#T93" onmouseover="rs('glibpre0/T93')" onmouseout="rh()">GLIBPRE0:93</a></span>
;<br/><a NAME="E23:108_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c15">w2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><a NAME="E24:108_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c15">w2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E25:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A11</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E26:108_1_2"/>
<font color="Maroon" title="c12">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c10">P1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_1_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><span class="kw">then </span>1 = 
<font color="Maroon" title="c12">w1</font> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font></span>)</span> <a href="glib_013.html#K5" title="GLIB_013:func.5">.degree()</a> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="glibpre0.html#T93" onmouseover="rs('glibpre0/T93')" onmouseout="rh()">GLIBPRE0:93</a></span>
;<br/><a NAME="E28:108_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c15">w2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:108_1_2"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="ref" href="glib_006.html#T56" onmouseover="rs('glib_006/T56')" onmouseout="rh()">GLIB_006:56</a>, <a class="ref" href="glib_000.html#T174" onmouseover="rs('glib_000/T174')" onmouseout="rh()">GLIB_000:174</a></span>;<br/><a NAME="E29:108_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c14">v2</font>,<font color="Maroon" title="c15">w2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="ref" href="partfun1.html#D6" onmouseover="rs('partfun1/D6')" onmouseout="rh()">PARTFUN1:def 6</a></span>;<br/><a NAME="E30:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A12</font></span>: 
( <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> or <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" onmouseover="rs('tarski/D2')" onmouseout="rh()">TARSKI:def 2</a></span>;<br/><a NAME="E31:108_1_2"/>
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">w1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="108_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:108_1_2_5"/>
<font color="Maroon" title="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w1</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"> contradiction</span><br/>

<a NAME="E2:108_1_2_5"/><span class="kw">then </span>
 <a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c10">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c11">v1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:108_1_2"><span class="lab"><font color="Green" title="E55">A5</font></span></a>, <a class="ref" href="enumset1.html#T29" onmouseover="rs('enumset1/T29')" onmouseout="rh()">ENUMSET1:29</a></span>;<br/>
<a NAME="E3:108_1_2_5"/><span class="kw">then </span>
 <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="glib_006.html#K6" title="GLIB_006:func.6">Endvertices</a> <font color="Maroon" title="c10">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="card_1.html#T30" onmouseover="rs('card_1/T30')" onmouseout="rh()">CARD_1:30</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:108_1_2_5"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glpacy00.html#T29" target="_self" onmouseover="rs('glpacy00/T29')" onmouseout="rh()">Th37</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><a NAME="E32:108_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A13</font></span>: 
( ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w2</font> ) or ( <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c11">v1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">w2</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c16">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="c12">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">v2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:108_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E25:108_1_2"><span class="lab"><font color="Green" title="E61">A11</font></span></a>, <a class="txt" href="#E30:108_1_2"><span class="lab"><font color="Green" title="E62">A12</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c2">C1</font>,<font color="Maroon" title="c3">C2</font><span class="kw"> such that </span><br/><a NAME="E34:108_1_2"/>
<font color="Maroon" title="c17">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c16">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c16">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p3">(<span class="default"> the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c2">C1</font> <a href="funcop_1.html#K17" title="FUNCOP_1:func.17">.--&gt;</a>  the   <a href="glib_008.html#NM1" title="GLIB_008:NM.1">Edge</a> of <font color="Maroon" title="c3">C2</font></span>)</span></span>)</span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">and </span><br/><a NAME="E35:108_1_2"/>
( <font color="Maroon" title="c16">F0</font> is  <a href="glib_010.html#V15" title="GLIB_010:attr.15">weak_SG-embedding</a>  implies <font color="Maroon" title="c17">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="E36:108_1_2"/><span class="lab"><font color="Green" title="E64">A14</font></span>: 
( <font color="Maroon" title="c16">F0</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  implies <font color="Maroon" title="c17">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="#E6:108_1_2"><span class="lab"><font color="Green" title="E54">A4</font></span></a>, <a class="txt" href="#E10:108_1_2"><span class="lab"><font color="Green" title="E56">A6</font></span></a>, <a class="txt" href="#E20:108_1_2"><span class="lab"><font color="Green" title="E60">A10</font></span></a>, <a class="txt" href="#E32:108_1_2"><span class="lab"><font color="Green" title="E63">A13</font></span></a>, <a class="ref" href="glib_010.html#T153" onmouseover="rs('glib_010/T153')" onmouseout="rh()">GLIB_010:153</a></span>;<br/><span class="kw">thus </span><a NAME="E37:108_1_2"/>
<font color="Maroon" title="c3">C2</font> is <font color="Maroon" title="c2">C1</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="#E19:108_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="txt" href="#E36:108_1_2"><span class="lab"><font color="Green" title="E64">A14</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><span class="kw">end;</span></div>

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