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<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">G3</font> being    <a href="glib_009.html#M5" title="GLIB_009:mode.5">SimpleGraph</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">G4</font> being    <a href="glib_009.html#M5" title="GLIB_009:mode.5">SimpleGraph</a> of <font color="Maroon" title="c2">G2</font><br/>  for <font color="Olive" title="b3">G5</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b4">G6</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b2">G4</font> is <font color="Olive" title="b1">G3</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  holds <br/><font color="Olive" title="b4">G6</font> is <font color="Olive" title="b3">G5</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="c3">G3</font> be    <a href="glib_009.html#M5" title="GLIB_009:mode.5">SimpleGraph</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">  for <font color="Olive" title="b1">G4</font> being    <a href="glib_009.html#M5" title="GLIB_009:mode.5">SimpleGraph</a> of <font color="Maroon" title="c2">G2</font><br/>  for <font color="Olive" title="b2">G5</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b3">G6</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">G4</font> is <font color="Maroon" title="c3">G3</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  holds <br/><font color="Olive" title="b3">G6</font> is <font color="Olive" title="b2">G5</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_009.html#M5" title="GLIB_009:mode.5">SimpleGraph</a> of <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">G5</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c1">G1</font><br/>  for <font color="Olive" title="b2">G6</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c4">G4</font> is <font color="Maroon" title="c3">G3</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  holds <br/><font color="Olive" title="b2">G6</font> is <font color="Olive" title="b1">G5</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">G5</font> be    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</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">  for <font color="Olive" title="b1">G6</font> being    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c2">G2</font>  st <font color="Maroon" title="c4">G4</font> is <font color="Maroon" title="c3">G3</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  holds <br/><font color="Olive" title="b1">G6</font> is <font color="Maroon" title="c5">G5</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">G6</font> be    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <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"> ( <font color="Maroon" title="c4">G4</font> is <font color="Maroon" title="c3">G3</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  implies <font color="Maroon" title="c6">G6</font> is <font color="Maroon" title="c5">G5</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a>  )</span><br/>









<a NAME="E1:164"/><span class="lab"><font color="Green" title="E79">A1</font></span>: 
( <font color="Maroon" title="c5">G5</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c3">G3</font> &amp; <font color="Maroon" title="c6">G6</font> is    <a href="glib_012.html#M5" title="GLIB_012:mode.5">GraphComplement</a> of <font color="Maroon" title="c4">G4</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_012.html#T101" target="_self" onmouseover="rs('glib_012/T101')" onmouseout="rh()">Th101</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:164"/>
<font color="Maroon" title="c4">G4</font> is <font color="Maroon" title="c3">G3</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"> <font color="Maroon" title="c6">G6</font> is <font color="Maroon" title="c5">G5</font> <a href="glib_010.html#V19" title="GLIB_010:attr.19">-isomorphic</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">f</font> being    <a href="glib_011.html#M1" title="GLIB_011:mode.1">PVertexMapping</a> of <font color="Maroon" title="c3">G3</font>,<font color="Maroon" title="c4">G4</font><span class="kw"> such that </span><br/><a NAME="E4:164"/><span class="lab"><font color="Green" title="E80">A2</font></span>: 
<font color="Maroon" title="c7">f</font> is  <a href="glib_011.html#V4" title="GLIB_011:attr.4">isomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_011.html#T49" onmouseover="rs('glib_011/T49')" onmouseout="rh()">GLIB_011:49</a></span>;<br/>
<a NAME="E5:164"/><span class="lab"><font color="Green" title="E81">A3</font></span>: 
(  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">G3</font> <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="c5">G5</font> &amp;  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c4">G4</font> <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="c6">G6</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">g</font> = <font color="Maroon" title="c7">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c5">G5</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c6">G6</font></span>)</span> ;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="164_1"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c5">G5</font> holds <br/> ex <font color="Olive" title="b4">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b4">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">w</font>,<font color="Maroon" title="c6">G6</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">w</font>, <font color="Maroon" title="c11">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c5">G5</font> implies  ex <font color="Olive" title="b1">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G6</font> )</span><br/><span class="kw">assume </span><a NAME="E1:164_1"/><span class="lab"><font color="Green" title="E82">A4</font></span>: 
( <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">g</font> &amp; <font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c5">G5</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">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G6</font></span><br/><a NAME="E2:164_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A5</font></span>: 
<font color="Maroon" title="c9">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c10">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</a></span>;<br/><a NAME="E3:164_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A6</font></span>: 
<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:164"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E1:164_1"><span class="lab"><font color="Green" title="E82">A4</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/><a NAME="E4:164_1"/><span class="lab"><font color="Green" title="E85">A7</font></span>: 
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c3">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E5:164"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E1:164_1"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E2:164_1"><span class="lab"><font color="Green" title="E83">A5</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">thus </span><a NAME="E5:164_1"/>
 ex <font color="Olive" title="b1">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G6</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="164_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:164_1_1"/><span class="lab"><font color="Green" title="E86">A8</font></span>: 
 for <font color="Olive" title="b1">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G6</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="E2:164_1_1"/>
( <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_1"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:164_1_1"/><span class="lab"><font color="Green" title="E87">A9</font></span>: 
<font color="Maroon" title="c12">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c8">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c4">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E3:164_1"><span class="lab"><font color="Green" title="E84">A6</font></span></a>, <a class="txt" href="#E1:164_1_1"><span class="lab"><font color="Green" title="E86">A8</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c13">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:164_1_1"/><span class="lab"><font color="Green" title="E88">A10</font></span>: 
<font color="Maroon" title="c13">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c3">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:164"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E1:164_1"><span class="lab"><font color="Green" title="E82">A4</font></span></a>, <a class="txt" href="#E4:164_1_1"><span class="lab"><font color="Green" title="E87">A9</font></span></a>, <a class="ref" href="glib_011.html#T2" onmouseover="rs('glib_011/T2')" onmouseout="rh()">GLIB_011:2</a></span>;<br/>
<span class="kw">thus </span><a NAME="E7:164_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:164_1"><span class="lab"><font color="Green" title="E85">A7</font></span></a>, <a class="txt" href="#E6:164_1_1"><span class="lab"><font color="Green" title="E88">A10</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/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">g</font> = <font color="Maroon" title="c8">g</font> as    <a href="glib_011.html#M1" title="GLIB_011:mode.1">PVertexMapping</a> of <font color="Maroon" title="c5">G5</font>,<font color="Maroon" title="c6">G6</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_011.html#T1" onmouseover="rs('glib_011/T1')" onmouseout="rh()">GLIB_011:1</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="164_2"><span class="kw">now </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">  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">w</font>, <font color="Olive" title="b3">e6</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Olive" title="b2">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Olive" title="b3">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">v</font>,<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">w</font>,<font color="Maroon" title="c6">G6</font> holds <br/> ex <font color="Olive" title="b4">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b4">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">w</font>,<font color="Maroon" title="c5">G5</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c10">v</font>, <font color="Maroon" title="c11">w</font>, <font color="Maroon" title="c12">e6</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</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="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c12">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c6">G6</font> implies  ex <font color="Olive" title="b1">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c5">G5</font> )</span><br/><span class="kw">assume </span><a NAME="E1:164_2"/><span class="lab"><font color="Green" title="E82">A11</font></span>: 
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c9">g</font> &amp; <font color="Maroon" title="c12">e6</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c6">G6</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">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c5">G5</font></span><br/><a NAME="E2:164_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A12</font></span>: 
<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T18" onmouseover="rs('glib_000/T18')" onmouseout="rh()">GLIB_000:18</a></span>;<br/><a NAME="E3:164_2"/><span class="lab"><font color="Green" title="E84">A13</font></span>: 
( <font color="Maroon" title="c10">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">G3</font> &amp; <font color="Maroon" title="c11">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K24" title="GLIB_000:func.24">the_Vertices_of</a> <font color="Maroon" title="c3">G3</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:164"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="txt" href="#E1:164_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a></span>;<br/><a NAME="E4:164_2"/>
( <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> &amp; <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c7">f</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><a NAME="E5:164_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A14</font></span>: 
 for <font color="Olive" title="b1">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c9">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c4">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E1:164_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E2:164_2"><span class="lab"><font color="Green" title="E83">A12</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/><span class="kw">thus </span><a NAME="E6:164_2"/>
 ex <font color="Olive" title="b1">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c5">G5</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"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="164_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:164_2_1"/>
 for <font color="Olive" title="b1">e5</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   holds  not <font color="Olive" title="b1">e5</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c5">G5</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/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:164_2_1"/><span class="lab"><font color="Green" title="E86">A15</font></span>: 
<font color="Maroon" title="c13">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c3">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164"><span class="lab"><font color="Green" title="E79">A1</font></span></a>, <a class="txt" href="#E2:164_2"><span class="lab"><font color="Green" title="E83">A12</font></span></a>, <a class="txt" href="#E3:164_2"><span class="lab"><font color="Green" title="E84">A13</font></span></a>, <a class="ref" href="glib_012.html#T98" target="_self" onmouseover="rs('glib_012/T98')" onmouseout="rh()">Th98</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">e4</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E5:164_2_1"/><span class="lab"><font color="Green" title="E87">A16</font></span>: 
<font color="Maroon" title="c14">e4</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">v</font>,<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">w</font>,<font color="Maroon" title="c4">G4</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:164_2"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E3:164_2_1"><span class="lab"><font color="Green" title="E86">A15</font></span></a>, <a class="ref" href="glib_011.html#T1" onmouseover="rs('glib_011/T1')" onmouseout="rh()">GLIB_011:1</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:164_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:164_2"><span class="lab"><font color="Green" title="E85">A14</font></span></a>, <a class="txt" href="#E5:164_2_1"><span class="lab"><font color="Green" title="E87">A16</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/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<a NAME="E10:164"/><span class="kw">then </span>
<font color="Maroon" title="c9">g</font> is  <a href="glib_011.html#V2" title="GLIB_011:attr.2">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_011.html#T2" onmouseover="rs('glib_011/T2')" onmouseout="rh()">GLIB_011:2</a></span>;<br/>
<span class="kw">hence </span><a NAME="E11:164"/>
<font color="Maroon" title="c6">G6</font> is <font color="Maroon" title="c5">G5</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="#E4:164"><span class="lab"><font color="Green" title="E80">A2</font></span></a>, <a class="txt" href="#E5:164"><span class="lab"><font color="Green" title="E81">A3</font></span></a>, <a class="ref" href="glib_011.html#T49" onmouseover="rs('glib_011/T49')" onmouseout="rh()">GLIB_011:49</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>
