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<span class="kw">let </span><font color="Maroon" title="c1">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">v</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <br/>  for <font color="Olive" title="b2">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b3">G1</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font>  st <font color="Olive" title="b2">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/> for <font color="Olive" title="b4">e1</font>, <font color="Olive" title="b5">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b5">u</font>,<font color="Olive" title="b3">G1</font> &amp; (  not <font color="Olive" title="b5">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">V</font> implies  not <font color="Olive" title="b4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b5">u</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">G1</font> ) &amp; (  for <font color="Olive" title="b6">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b4">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b5">u</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">G1</font> &amp; <font color="Olive" title="b6">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b5">u</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b3">G1</font> holds <br/><font color="Olive" title="b4">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">e2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">v</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">  for <font color="Olive" title="b1">V</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <br/>  for <font color="Olive" title="b2">G1</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font>  st <font color="Olive" title="b1">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/> for <font color="Olive" title="b3">e1</font>, <font color="Olive" title="b4">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b4">u</font>,<font color="Olive" title="b2">G1</font> &amp; (  not <font color="Olive" title="b4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">V</font> implies  not <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b4">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">G1</font> ) &amp; (  for <font color="Olive" title="b5">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b4">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">G1</font> &amp; <font color="Olive" title="b5">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b4">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">G1</font> holds <br/><font color="Olive" title="b3">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">e2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">V</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">G1</font> being    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>  st <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> holds <br/> for <font color="Olive" title="b2">e1</font>, <font color="Olive" title="b3">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b3">u</font>,<font color="Olive" title="b1">G1</font> &amp; (  not <font color="Olive" title="b3">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b3">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">G1</font> ) &amp; (  for <font color="Olive" title="b4">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b3">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">G1</font> &amp; <font color="Olive" title="b4">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b3">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">G1</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">e2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font> be    <a href="glib_007.html#M3" title="GLIB_007:mode.3">addAdjVertexFromAll</a> of <font color="Maroon" title="c1">G2</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> implies  for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">u</font>,<font color="Maroon" title="c4">G1</font> &amp; (  not <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> ) &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) ) )</span><br/>







<span class="kw">assume </span><a NAME="E1:65"/><span class="lab"><font color="Green" title="E39">A1</font></span>: 
( <font color="Maroon" title="c3">V</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">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">e1</font>, <font color="Olive" title="b2">u</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  holds <br/> (  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Olive" title="b2">u</font>,<font color="Maroon" title="c4">G1</font> &amp; (  not <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> ) &amp; (  for <font color="Olive" title="b3">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Olive" title="b3">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )</span><br/>

<a NAME="E2:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E40">A2</font></span>: 
(  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">v</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span>)</span> &amp;  <a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c1">G2</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp;  <a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">G2</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D3" target="_self" onmouseover="rs('glib_007/D3')" onmouseout="rh()">Def3</a></span>;<br/>
<span class="kw">let </span><font color="Maroon" title="c5">e1</font>, <font color="Maroon" title="c6">u</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"> (  not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c4">G1</font> &amp; (  not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> ) &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font> ) )</span><br/>

<span class="kw">thus </span><a NAME="E3:65"/>
 not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c4">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"> ( (  not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> ) &amp; (  for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font> ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_1"/>
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c4">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"> contradiction</span><br/>

<a NAME="E2:65_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A3</font></span>: 
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</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/>
<a NAME="E3:65_1"/>
 not <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">reconsider </span><font color="Maroon" title="c7">e3</font> = <font color="Maroon" title="c5">e1</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:65_1_1"/><span class="lab"><font color="Green" title="E42">A4</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">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"> contradiction</span><br/>

<a NAME="E3:65_1_1"/><span class="kw">then </span>
<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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:65_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E2:65_1_1"><span class="lab"><font color="Green" title="E42">A4</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</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="E4:65_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:65_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E5:65_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
;<br/>
<a NAME="E6:65_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c5">e1</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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">x1</font>, <font color="Maroon" title="c8">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E8:65_1"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
( <font color="Maroon" title="c7">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c8">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">and </span><br/><a NAME="E9:65_1"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c5">e1</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"><font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:65_1"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<font color="Maroon" title="c2">v</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E2:65_1"><span class="lab"><font color="Green" title="E41">A3</font></span></a>, <a class="txt" href="#E6:65_1"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x1</font>,<font color="Maroon" title="c8">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:65_1"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65_1"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E11:65_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E8:65_1"><span class="lab"><font color="Green" title="E44">A7</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>

<span class="kw">thus </span><a NAME="E4:65"/>
(  not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> implies  not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">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">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font></span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_2"/><span class="lab"><font color="Green" title="E41">A9</font></span>: 
 not <font color="Maroon" title="c6">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  not <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font></span><br/>

<span class="kw">assume </span><a NAME="E2:65_2"/>
<font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">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"> contradiction</span><br/>

<a NAME="E3:65_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A10</font></span>: 
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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/>
<a NAME="E4:65_2"/>
 not <font color="Maroon" title="c5">e1</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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_2_1"/><span class="lab"><font color="Green" title="E43">A11</font></span>: 
<font color="Maroon" title="c5">e1</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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</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"> contradiction</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">x</font>, <font color="Maroon" title="c8">y</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:65_2_1"/><span class="lab"><font color="Green" title="E44">A12</font></span>: 
( <font color="Maroon" title="c7">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c8">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; <font color="Maroon" title="c5">e1</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"><font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<font color="Maroon" title="c6">u</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:65_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E1:65_2_1"><span class="lab"><font color="Green" title="E43">A11</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">y</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_2_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c7">x</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:65_2_1"><span class="lab"><font color="Green" title="E44">A12</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:65_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65_2"><span class="lab"><font color="Green" title="E41">A9</font></span></a>, <a class="txt" href="#E3:65_2_1"><span class="lab"><font color="Green" title="E44">A12</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>
<a NAME="E5:65_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<a NAME="E6:65_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A13</font></span>: 
 not <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span>
 
;<br/>
<a NAME="E7:65_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c5">e1</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> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span>)</span> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <font color="Maroon" title="c2">v</font></span>)</span>
 
;<br/>
<a NAME="E8:65_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A14</font></span>: 
<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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>;<br/>
<a NAME="E9:65_2"/>
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E3:65_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E6:65_2"><span class="lab"><font color="Green" title="E43">A13</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:65_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:65_2"><span class="lab"><font color="Green" title="E44">A14</font></span></a>, <a class="txt" href="#E3:65_2"><span class="lab"><font color="Green" title="E42">A10</font></span></a>, <a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</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>

<span class="kw">let </span><font color="Maroon" title="c7">e2</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="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> implies <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E5:65"/>
( <font color="Maroon" title="c5">e1</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">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="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font></span><br/>

<a NAME="E6:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E41">A15</font></span>: 
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> &amp; <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">u</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font> &amp; <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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</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/>
<a NAME="E7:65"/>
(  not <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> &amp;  not <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> )
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="65_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:65_3"/>
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> or <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">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"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_3_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="E1:65_3_1"/><span class="kw">then </span>
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> or <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:65_3_1_1"/><span class="lab"><font color="Green" title="E42">A16</font></span>: 
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">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"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">e3</font> = <font color="Maroon" title="c5">e1</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E3:65_3_1_1"/>
<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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E1:65_3_1_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:65_3_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:65_3_1_1"><span class="lab"><font color="Green" title="E42">A16</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="65_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:65_3_1_2"/><span class="lab"><font color="Green" title="E42">A17</font></span>: 
<font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">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"> contradiction</span><br/><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c8">e3</font> = <font color="Maroon" title="c7">e2</font> as    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#T1" onmouseover="rs('tarski/T1')" onmouseout="rh()">TARSKI:1</a></span>;<br/><a NAME="E3:65_3_1_2"/>
<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="c1">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">v</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E1:65_3_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/><span class="kw">hence </span><a NAME="E4:65_3_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:65"><span class="lab"><font color="Green" title="E39">A1</font></span></a>, <a class="txt" href="#E1:65_3_1_2"><span class="lab"><font color="Green" title="E42">A17</font></span></a>, <a class="ref" href="funct_2.html#T5" onmouseover="rs('funct_2/T5')" onmouseout="rh()">FUNCT_2:5</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>

</div><span class="kw">end;</span></div>
<a NAME="E8:65"/><span class="kw">then </span>
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> &amp; <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> <a href="funcop_1.html#K7" title="FUNCOP_1:func.7">--&gt;</a> <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="ref" href="xboole_0.html#D3" onmouseover="rs('xboole_0/D3')" onmouseout="rh()">XBOOLE_0:def 3</a></span>;<br/>
<a NAME="E9:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E42">A18</font></span>: 
( <font color="Maroon" title="c5">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; <font color="Maroon" title="c7">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> )
 
;<br/>
<a NAME="E10:65"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
( <font color="Maroon" title="c5">e1</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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> &amp; <font color="Maroon" title="c7">e2</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> <span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c8">x1</font>, <font color="Maroon" title="c9">y1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E12:65"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
( <font color="Maroon" title="c8">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c9">y1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">and </span><br/><a NAME="E13:65"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
<font color="Maroon" title="c5">e1</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"><font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c10">x2</font>, <font color="Maroon" title="c11">y2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E15:65"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
( <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> &amp; <font color="Maroon" title="c11">y2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 <span class="kw">and </span><br/><a NAME="E16:65"/><span class="lab"><font color="Green" title="E47">A23</font></span>: 
<font color="Maroon" title="c7">e2</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"><font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:65"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="ref" href="zfmisc_1.html#D2" onmouseover="rs('zfmisc_1/D2')" onmouseout="rh()">ZFMISC_1:def 2</a></span>;<br/>
<span class="lab"><font color="Green" title="E48">A24</font></span>: <font color="Maroon" title="c6">u</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">e1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c8">x1</font>,<font color="Maroon" title="c9">y1</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c8">x1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>
;<br/>
<span class="lab"><font color="Green" title="E49">A25</font></span>: <font color="Maroon" title="c6">u</font> = 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">e2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:65"><span class="lab"><font color="Green" title="E40">A2</font></span></a>, <a class="txt" href="#E6:65"><span class="lab"><font color="Green" title="E41">A15</font></span></a>, <a class="txt" href="#E10:65"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="funct_3.html#K9" title="FUNCT_3:func.9">pr1</a> (<font color="Maroon" title="c3">V</font>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p3">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G2</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>)</span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c10">x2</font>,<font color="Maroon" title="c11">y2</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:65"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="binop_1.html#D1" onmouseover="rs('binop_1/D1')" onmouseout="rh()">BINOP_1:def 1</a></span>
<br/>.= 
<font color="Maroon" title="c10">x2</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="funct_3.html#D4" onmouseover="rs('funct_3/D4')" onmouseout="rh()">FUNCT_3:def 4</a></span>
;<br/>
<a NAME="E19:65"/>
( <font color="Maroon" title="c9">y1</font> <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">G2</font> &amp; <font color="Maroon" title="c11">y2</font> <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">G2</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:65"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E15:65"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E20:65"/>
<font color="Maroon" title="c5">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:65"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E16:65"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="txt" href="#E17:65"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="txt" href="#E18:65"><span class="lab"><font color="Green" title="E49">A25</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>
