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<span class="kw">let </span><font color="Maroon" title="c1">G</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>, <font color="Olive" title="b4">G2</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Olive" title="b1">v</font>,<font color="Olive" title="b2">V</font>  ex <font color="Olive" title="b5">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b3">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b4">G2</font></span>)</span> st <br/>( <font color="Olive" title="b5">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b5">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b5">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b6">v1</font>, <font color="Olive" title="b7">e</font>, <font color="Olive" title="b8">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b8">v2</font>,<font color="Olive" title="b3">G1</font> holds <br/><font color="Olive" title="b5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b7">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b6">v1</font>,<font color="Olive" title="b8">v2</font>,<font color="Olive" title="b4">G2</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>, <font color="Olive" title="b3">G2</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Olive" title="b1">V</font>  ex <font color="Olive" title="b4">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b2">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b3">G2</font></span>)</span> st <br/>( <font color="Olive" title="b4">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b4">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b4">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b5">v1</font>, <font color="Olive" title="b6">e</font>, <font color="Olive" title="b7">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v2</font>,<font color="Olive" title="b2">G1</font> holds <br/><font color="Olive" title="b4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b6">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b5">v1</font>,<font color="Olive" title="b7">v2</font>,<font color="Olive" title="b3">G2</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>, <font color="Olive" title="b2">G2</font> being    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c3">V</font>  ex <font color="Olive" title="b3">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b1">G1</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Olive" title="b2">G2</font></span>)</span> st <br/>( <font color="Olive" title="b3">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b3">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b3">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b4">v1</font>, <font color="Olive" title="b5">e</font>, <font color="Olive" title="b6">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b1">G1</font> holds <br/><font color="Olive" title="b3">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b5">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b4">v1</font>,<font color="Olive" title="b6">v2</font>,<font color="Olive" title="b2">G2</font> ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">G1</font>, <font color="Maroon" title="c5">G2</font> be    <a href="glib_007.html#M4" title="GLIB_007:mode.4">addAdjVertexAll</a> of <font color="Maroon" title="c1">G</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">  ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">e</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/>







<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_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="E1:101_1"/>
( ( <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">G</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">G</font> ) or  not <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">G</font> or <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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1"/><span class="lab"><font color="Green" title="E65">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">G</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">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">e</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><div class="add"><span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ] <span class="kw">means </span> for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st $1 <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/>$2 <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b2">v2</font>,<font color="Maroon" title="c5">G2</font>;<br/><a NAME="E2:101_1_1"/><span class="lab"><font color="Green" title="E66">A2</font></span>: 
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> holds <br/> ex <font color="Olive" title="b2">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">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="c5">G2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e1</font>,<font color="Olive" title="b2">e2</font>] )
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c6">e1</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="c6">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> implies  ex <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">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="c5">G2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">e1</font>,<font color="Olive" title="b1">e2</font>] ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_1_1"/>
<font color="Maroon" title="c6">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>
 ; <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">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b1">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="c5">G2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">e1</font>,<font color="Olive" title="b1">e2</font>] )</span><br/>

<a NAME="E2:101_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c6">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c4">G1</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c7">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E4:101_1_1_1"/><span class="lab"><font color="Green" title="E67">A3</font></span>: 
<font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#T67" target="_self" onmouseover="rs('glib_007/T67')" onmouseout="rh()">Th67</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c7">e2</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="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="c5">G2</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c7">e2</font>] )</span><br/>

<span class="kw">thus </span><a NAME="E5:101_1_1_1"/>
<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="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_1"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</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"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">e1</font>,<font color="Maroon" title="c7">e2</font>]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">v2</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="c6">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c4">G1</font> implies <font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E6:101_1_1_1"/>
<font color="Maroon" title="c6">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</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="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_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="E1:101_1_1_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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</font> ) or ( <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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_1_1_1"/>
( <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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</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="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_1_1_1_1_1"/>
<font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_1"><span class="lab"><font color="Green" title="E67">A3</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_1_1_2"/>
( <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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">v2</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="c6">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">v1</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="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:101_1_1_1_1_2"/>
<font color="Maroon" title="c7">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c9">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_1"><span class="lab"><font color="Green" title="E67">A3</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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><span class="kw">consider </span><font color="Maroon" title="c6">f1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E4:101_1_1"/><span class="lab"><font color="Green" title="E67">A4</font></span>: 
 for <font color="Olive" title="b1">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e1</font>,<font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e1</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S1" onmouseover="rs('funct_2/S1')" onmouseout="rh()">FUNCT_2:sch 1</a>(<a class="txt" href="#E2:101_1_1"><span class="lab"><font color="Green" title="E66">A2</font></span></a>);<br/></span><span class="kw">set </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c6">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>;<br/><a NAME="E5:101_1_1"/><span class="lab"><font color="Green" title="E68">A5</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/><a NAME="E6:101_1_1"/><span class="lab"><font color="Green" title="E69">A6</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2"><span class="kw">proof </span></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_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="E1:101_1_1_2_1"/>
( <font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or ( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_1"/>
<font color="Maroon" title="c3">V</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font></span><br/><div class="add"><a NAME="E2:101_1_1_2_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="glib_007.html#T47" target="_self" onmouseover="rs('glib_007/T47')" onmouseout="rh()">Th47</a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_1_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</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="101_1_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_2"/><span class="lab"><font color="Green" title="E70">A7</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font></span><br/><div class="add"><a NAME="E2:101_1_1_2_1_2"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_2_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:101_1_1_2_1_2_1"/><span class="lab"><font color="Green" title="E71">A8</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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"> contradiction</span><br/>

<a NAME="E2:101_1_1_2_1_2_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font>
 
<span class="kw">by</span> <span class="lab"><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="E3:101_1_1_2_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_2_1_2"><span class="lab"><font color="Green" title="E70">A7</font></span></a>, <a class="txt" href="#E1:101_1_1_2_1_2_1"><span class="lab"><font color="Green" title="E71">A8</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">hence </span><a NAME="E3:101_1_1_2_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</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="101_1_1_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_2_1_3"/>
( <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</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">  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="c4">G1</font></span><br/><div class="add"><a NAME="E2:101_1_1_2_1_3"/><span class="kw">then </span>
(  <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>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="glib_007.html#E61"><span class="lab"><font color="Green" title="E48">Lm11</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E3:101_1_1_2_1_3"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</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="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"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="lab"><font color="Green" title="E70">A9</font></span>:  <a href="relat_1.html#NK1" title="RELAT_1:NK.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#D1" onmouseover="rs('funct_4/D1')" onmouseout="rh()">FUNCT_4:def 1</a></span>
<br/>.= 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1"><span class="lab"><font color="Green" title="E68">A5</font></span></a>, <a class="txt" href="#E6:101_1_1"><span class="lab"><font color="Green" title="E69">A6</font></span></a>, <a class="ref" href="xboole_1.html#T12" onmouseover="rs('xboole_1/T12')" onmouseout="rh()">XBOOLE_1:12</a></span>
;<br/><a NAME="E8:101_1_1"/><span class="lab"><font color="Green" title="E71">A10</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_006.html#D9" onmouseover="rs('glib_006/D9')" onmouseout="rh()">GLIB_006:def 9</a></span>;<br/><a NAME="E9:101_1_1"/><span class="lab"><font color="Green" title="E72">A11</font></span>: 
<span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_1"><span class="lab"><font color="Green" title="E71">A10</font></span></a>, <a class="ref" href="xboole_1.html#T8" onmouseover="rs('xboole_1/T8')" onmouseout="rh()">XBOOLE_1:8</a></span>;<br/><a NAME="E10:101_1_1"/>
 <a href="xtuple_0.html#K10" title="XTUPLE_0:func.10">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">f1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T17" onmouseover="rs('funct_4/T17')" onmouseout="rh()">FUNCT_4:17</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c6">f1</font> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="funct_2.html#T2" onmouseover="rs('funct_2/T2')" onmouseout="rh()">FUNCT_2:2</a>, <a class="txt" href="#E9:101_1_1"><span class="lab"><font color="Green" title="E72">A11</font></span></a>, <a class="ref" href="xboole_1.html#T1" onmouseover="rs('xboole_1/T1')" onmouseout="rh()">XBOOLE_1:1</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c8">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><span class="kw">thus </span><font color="Maroon" title="c8">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> = 
<font color="Maroon" title="c8">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p2">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
 <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></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"> ( <font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c8">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c9">E1</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E14:101_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c9">E1</font>
 <span class="kw">and </span><br/><a NAME="E15:101_1_1"/><span class="lab"><font color="Green" title="E73">A12</font></span>: 
( <font color="Maroon" title="c9">E1</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &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">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c9">E1</font> )
 <span class="kw">and </span><br/><a NAME="E16:101_1_1"/><span class="lab"><font color="Green" title="E74">A13</font></span>: 
 for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</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="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">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="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c10">E2</font> being    <a href="hidden.html#M2" title="HIDDEN:mode.2">set</a> <span class="kw"> such that </span><br/><a NAME="E18:101_1_1"/>
 <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c3">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K1" title="CARD_1:func.1">card</a> <font color="Maroon" title="c10">E2</font>
 <span class="kw">and </span><br/><a NAME="E19:101_1_1"/><span class="lab"><font color="Green" title="E75">A14</font></span>: 
( <font color="Maroon" title="c10">E2</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</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">G</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <font color="Maroon" title="c10">E2</font> )
 <span class="kw">and </span><br/><a NAME="E20:101_1_1"/><span class="lab"><font color="Green" title="E76">A15</font></span>: 
 for <font color="Olive" title="b1">v1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">v1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">V</font> holds <br/> ex <font color="Olive" title="b2">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <br/>( <font color="Olive" title="b2">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E2</font> &amp; <font color="Olive" title="b2">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</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="b3">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font> holds <br/><font color="Olive" title="b2">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">e2</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E21:101_1_1"/>
 for <font color="Olive" title="b1">e1</font>, <font color="Olive" title="b2">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">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> <font color="Maroon" title="c8">f</font> &amp; <font color="Olive" title="b2">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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e2</font> holds <br/><font color="Olive" title="b1">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">e1</font>, <font color="Maroon" title="c12">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="c11">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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c12">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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> implies <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:101_1_1_5"/><span class="lab"><font color="Green" title="E77">A16</font></span>: 
( <font color="Maroon" title="c11">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> <font color="Maroon" title="c8">f</font> &amp; <font color="Maroon" title="c12">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> <font color="Maroon" title="c8">f</font> )
 <span class="kw">and </span><br/><a NAME="E2:101_1_1_5"/><span class="lab"><font color="Green" title="E78">A17</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/>

<span class="kw">set </span><font color="Maroon" title="c13">x1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c14">y1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c15">x2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c16">y2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>;<br/>
<a NAME="E3:101_1_1_5"/><span class="lab"><font color="Green" title="E79">A18</font></span>: 
( <font color="Maroon" title="c11">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="c12">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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5"><span class="lab"><font color="Green" title="E77">A16</font></span></a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_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="E1:101_1_1_5_1"/>
( ( <font color="Maroon" title="c11">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">G</font> &amp; <font color="Maroon" title="c12">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">G</font> ) or (  not <font color="Maroon" title="c11">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">G</font> &amp; <font color="Maroon" title="c12">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">G</font> ) or (  not <font color="Maroon" title="c12">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">G</font> &amp; <font color="Maroon" title="c11">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">G</font> ) or (  not <font color="Maroon" title="c11">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">G</font> &amp;  not <font color="Maroon" title="c12">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">G</font> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_1"/><span class="lab"><font color="Green" title="E80">A19</font></span>: 
( <font color="Maroon" title="c11">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">G</font> &amp; <font color="Maroon" title="c12">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">G</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><a NAME="E2:101_1_1_5_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E3:101_1_1_5_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/><a NAME="E4:101_1_1_5_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5_1_1"><span class="lab"><font color="Green" title="E80">A19</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E5:101_1_1_5_1_1"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_5"><span class="lab"><font color="Green" title="E78">A17</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_2"/><span class="lab"><font color="Green" title="E80">A20</font></span>: 
(  not <font color="Maroon" title="c11">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">G</font> &amp; <font color="Maroon" title="c12">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">G</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><a NAME="E2:101_1_1_5_1_2"/><span class="kw">then </span>
(  not <font color="Maroon" title="c11">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c12">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E3:101_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A21</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/><a NAME="E4:101_1_1_5_1_2"/><span class="lab"><font color="Green" title="E82">A22</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E5:101_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A23</font></span>: 
(  not <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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> or  not <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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E15:101_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="glib_007.html#T51" target="_self" onmouseover="rs('glib_007/T51')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E6:101_1_1_5_1_2"/>
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E7:101_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A24</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><a NAME="E8:101_1_1_5_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A25</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</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="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><a NAME="E9:101_1_1_5_1_2"/><span class="lab"><font color="Green" title="E86">A26</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E4:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E82">A22</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><a NAME="E10:101_1_1_5_1_2"/><span class="lab"><font color="Green" title="E87">A27</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E84">A24</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</a></span>;<br/><a NAME="E11:101_1_1_5_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E80">A20</font></span></a>, <a class="txt" href="#E3:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E81">A21</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/><span class="kw">hence </span><a NAME="E12:101_1_1_5_1_2"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E83">A23</font></span></a>, <a class="txt" href="#E8:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E85">A25</font></span></a>, <a class="txt" href="#E9:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E86">A26</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a>, <a class="txt" href="#E2:101_1_1_5"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="txt" href="#E10:101_1_1_5_1_2"><span class="lab"><font color="Green" title="E87">A27</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_3"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_3"/><span class="lab"><font color="Green" title="E80">A28</font></span>: 
(  not <font color="Maroon" title="c12">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">G</font> &amp; <font color="Maroon" title="c11">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">G</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><a NAME="E2:101_1_1_5_1_3"/><span class="kw">then </span>
(  not <font color="Maroon" title="c12">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> &amp; <font color="Maroon" title="c11">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E3:101_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A29</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a>, <a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>;<br/><a NAME="E4:101_1_1_5_1_3"/><span class="lab"><font color="Green" title="E82">A30</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E5:101_1_1_5_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</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="c12">e2</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="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <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="c12">e2</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#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="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E15:101_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E80">A28</font></span></a>, <a class="ref" href="glib_007.html#T51" target="_self" onmouseover="rs('glib_007/T51')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E6:101_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A31</font></span>: 
(  not <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="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> or  not <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="c12">e2</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a></span>;<br/><a NAME="E7:101_1_1_5_1_3"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E8:101_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A32</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E80">A28</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><a NAME="E9:101_1_1_5_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A33</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#T13" onmouseover="rs('glib_000/T13')" onmouseout="rh()">GLIB_000:13</a></span>;<br/><a NAME="E10:101_1_1_5_1_3"/><span class="lab"><font color="Green" title="E86">A34</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E4:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E82">A30</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><a NAME="E11:101_1_1_5_1_3"/><span class="lab"><font color="Green" title="E87">A35</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E84">A32</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</a></span>;<br/><a NAME="E12:101_1_1_5_1_3"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E80">A28</font></span></a>, <a class="txt" href="#E3:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E81">A29</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>;<br/><a NAME="E13:101_1_1_5_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_5"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="txt" href="#E11:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E87">A35</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E14:101_1_1_5_1_3"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E83">A31</font></span></a>, <a class="txt" href="#E9:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E85">A33</font></span></a>, <a class="txt" href="#E10:101_1_1_5_1_3"><span class="lab"><font color="Green" title="E86">A34</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</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="101_1_1_5_1_4"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4"/><span class="lab"><font color="Green" title="E80">A36</font></span>: 
(  not <font color="Maroon" title="c11">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">G</font> &amp;  not <font color="Maroon" title="c12">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">G</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><a NAME="E2:101_1_1_5_1_4"/><span class="kw">then </span>
(  not <font color="Maroon" title="c11">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> &amp;  not <font color="Maroon" title="c12">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> )
 ;<br/><a NAME="E3:101_1_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A37</font></span>: 
( <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> &amp; <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>;<br/><a NAME="E4:101_1_1_5_1_4"/><span class="lab"><font color="Green" title="E82">A38</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E5:101_1_1_5_1_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A39</font></span>: 
( <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</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="c11">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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <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="c11">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#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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E15:101_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E80">A36</font></span></a>, <a class="ref" href="glib_007.html#T51" target="_self" onmouseover="rs('glib_007/T51')" onmouseout="rh()">Th51</a></span>;<br/><a NAME="E6:101_1_1_5_1_4"/><span class="lab"><font color="Green" title="E84">A40</font></span>: 
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E7:101_1_1_5_1_4"/><span class="lab"><font color="Green" title="E85">A41</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E4:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E3:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E81">A37</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><a NAME="E8:101_1_1_5_1_4"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_5"><span class="lab"><font color="Green" title="E79">A18</font></span></a>, <a class="txt" href="#E6:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="txt" href="#E3:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E81">A37</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_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="E1:101_1_1_5_1_4_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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> ) or ( <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="c11">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">e1</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_1_5"><span class="lab"><font color="Green" title="E78">A17</font></span></a>, <a class="txt" href="#E7:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E85">A41</font></span></a>, <a class="ref" href="glib_000.html#T15" onmouseover="rs('glib_000/T15')" onmouseout="rh()">GLIB_000:15</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_1"/><span class="lab"><font color="Green" title="E86">A42</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_1_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="E1:101_1_1_5_1_4_1_1_1"/>
( ( <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="c11">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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <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="c11">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#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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E83">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_1_1_1"/><span class="lab"><font color="Green" title="E87">A43</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_5_1_4_1_1_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_5_1_4_1_1_1_1"/><span class="lab"><font color="Green" title="E88">A44</font></span>: 
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_5_1_4_1_1_1_1"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="txt" href="#E4:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E87">A43</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">hence </span><a NAME="E6:101_1_1_5_1_4_1_1_1_1"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E88">A44</font></span></a>, <a class="txt" href="#E6:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E86">A42</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1_1_1"><span class="lab"><font color="Green" title="E87">A43</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</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="101_1_1_5_1_4_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E87">A46</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="c11">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#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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_5_1_4_1_1_1_2"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_5_1_4_1_1_1_2"/><span class="lab"><font color="Green" title="E88">A47</font></span>: 
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_5_1_4_1_1_1_2"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E87">A46</font></span></a></span>;<br/><a NAME="E6:101_1_1_5_1_4_1_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E88">A47</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:101_1_1_5_1_4_1_1_1_2"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E88">A47</font></span></a>, <a class="txt" href="#E6:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1"><span class="lab"><font color="Green" title="E86">A42</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_1_1_2"><span class="lab"><font color="Green" title="E87">A46</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></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_2"/><span class="lab"><font color="Green" title="E86">A49</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">e1</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e2</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="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_2_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="E1:101_1_1_5_1_4_1_2_1"/>
( ( <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="c11">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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <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="c11">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#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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E83">A39</font></span></a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_2_1_1"/><span class="lab"><font color="Green" title="E87">A50</font></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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_5_1_4_1_2_1_1"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_5_1_4_1_2_1_1"/><span class="lab"><font color="Green" title="E88">A51</font></span>: 
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_5_1_4_1_2_1_1"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E88">A51</font></span></a>, <a class="txt" href="#E4:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E87">A50</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">hence </span><a NAME="E6:101_1_1_5_1_4_1_2_1_1"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E88">A51</font></span></a>, <a class="txt" href="#E6:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2"><span class="lab"><font color="Green" title="E86">A49</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2_1_1"><span class="lab"><font color="Green" title="E87">A50</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_5_1_4_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_5_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E87">A53</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="c11">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#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="c11">e1</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_5_1_4_1_2_1_2"/>
( <font color="Maroon" title="c17">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c17">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_5_1_4_1_2_1_2"/><span class="lab"><font color="Green" title="E88">A54</font></span>: 
 for <font color="Olive" title="b1">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c17">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_5_1_4_1_2_1_2"/>
<font color="Maroon" title="c11">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E82">A38</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E87">A53</font></span></a></span>;<br/><a NAME="E6:101_1_1_5_1_4_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A55</font></span>: 
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E88">A54</font></span></a></span>;<br/><a NAME="E7:101_1_1_5_1_4_1_2_1_2"/>
<font color="Maroon" title="c12">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c4">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e1</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_5_1_4"><span class="lab"><font color="Green" title="E84">A40</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2"><span class="lab"><font color="Green" title="E86">A49</font></span></a>, <a class="txt" href="#E1:101_1_1_5_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E87">A53</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">hence </span><a NAME="E8:101_1_1_5_1_4_1_2_1_2"/>
<font color="Maroon" title="c11">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:101_1_1_5_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E88">A54</font></span></a>, <a class="txt" href="#E6:101_1_1_5_1_4_1_2_1_2"><span class="lab"><font color="Green" title="E89">A55</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></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E22:101_1_1"/>
<font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</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"> ( <font color="Maroon" title="c8">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><a NAME="E23:101_1_1"/>
 for <font color="Olive" title="b1">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e</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="c5">G2</font> holds <br/><font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="101_1_1_6"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c11">e</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="c5">G2</font> implies <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:101_1_1_6"/><span class="lab"><font color="Green" title="E77">A56</font></span>: 
<font color="Maroon" title="c11">e</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="c5">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="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_6_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="E1:101_1_1_6_1"/>
( <font color="Maroon" title="c11">e</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">G</font> or  not <font color="Maroon" title="c11">e</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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_6_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_6_1_1"/><span class="lab"><font color="Green" title="E78">A57</font></span>: 
<font color="Maroon" title="c11">e</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">G</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="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span><br/><div class="add"><a NAME="E2:101_1_1_6_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c11">e</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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A58</font></span>: <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c11">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_6_1_1"><span class="lab"><font color="Green" title="E78">A57</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/><a NAME="E4:101_1_1_6_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c4">G1</font>
 <span class="kw">by</span> <span class="lab"><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="E5:101_1_1_6_1_1"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_1"><span class="lab"><font color="Green" title="E79">A58</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a>, <a class="txt" href="#E7:101_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_1"><span class="lab"><font color="Green" title="E78">A57</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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_6_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_6_1_2"/><span class="lab"><font color="Green" title="E78">A59</font></span>: 
 not <font color="Maroon" title="c11">e</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">G</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="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c12">v1</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>;<br/><span class="kw">set </span><font color="Maroon" title="c13">v2</font> = <span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>;<br/><a NAME="E2:101_1_1_6_1_2"/><span class="lab"><font color="Green" title="E79">A60</font></span>: 
<font color="Maroon" title="c11">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<span class="p1">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_6"><span class="lab"><font color="Green" title="E77">A56</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_6_1_2_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="E1:101_1_1_6_1_2_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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) or ( <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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</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#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c3">V</font> ) )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1"><span class="lab"><font color="Green" title="E65">A1</font></span></a>, <a class="txt" href="#E19:101_1_1"><span class="lab"><font color="Green" title="E75">A14</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_2"><span class="lab"><font color="Green" title="E78">A59</font></span></a>, <a class="ref" href="glib_007.html#T51" target="_self" onmouseover="rs('glib_007/T51')" onmouseout="rh()">Th51</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_6_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E80">A61</font></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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E81">A62</font></span>: 
( <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_6_1_2_1_1"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E82">A63</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E6:101_1_1_6_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A64</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><a NAME="E7:101_1_1_6_1_2_1_1"/>
<font color="Maroon" title="c9">E1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:101_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>;<br/><a NAME="E8:101_1_1_6_1_2_1_1"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E81">A62</font></span></a>, <a class="txt" href="glib_007.html#E35"><span class="lab"><font color="Green" title="E30">Lm7</font></span></a></span>;<br/><a NAME="E9:101_1_1_6_1_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A65</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E83">A64</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:101_1_1_6_1_2_1_1"/>
( <font color="Maroon" title="c15">e3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E2</font> &amp; <font color="Maroon" title="c15">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font> )
 <span class="kw">and </span><br/><a NAME="E12:101_1_1_6_1_2_1_1"/><span class="lab"><font color="Green" title="E85">A66</font></span>: 
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font> holds <br/><font color="Maroon" title="c15">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:101_1_1"><span class="lab"><font color="Green" title="E76">A15</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E80">A61</font></span></a></span>;<br/><a NAME="E13:101_1_1_6_1_2_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E84">A65</font></span></a>, <a class="txt" href="#E12:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E85">A66</font></span></a></span>;<br/><a NAME="E14:101_1_1_6_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E85">A66</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E80">A61</font></span></a>, <a class="txt" href="#E2:101_1_1_6_1_2"><span class="lab"><font color="Green" title="E79">A60</font></span></a>, <a class="ref" href="glib_000.html#T14" onmouseover="rs('glib_000/T14')" onmouseout="rh()">GLIB_000:14</a></span>;<br/><span class="kw">hence </span><a NAME="E15:101_1_1_6_1_2_1_1"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_6_1_2_1_1"><span class="lab"><font color="Green" title="E82">A63</font></span></a>, <a class="txt" href="#E7:101_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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="101_1_1_6_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E80">A68</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</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#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font> <a href="tarski.html#R2" title="TARSKI: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"> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">e1</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:101_1_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E81">A69</font></span>: 
( <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">E1</font> &amp; <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> )
 <span class="kw">and </span><br/><a NAME="E4:101_1_1_6_1_2_1_2"/>
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:101_1_1"><span class="lab"><font color="Green" title="E74">A13</font></span></a></span>;<br/><a NAME="E5:101_1_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E82">A70</font></span>: 
<font color="Maroon" title="c14">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><a NAME="E6:101_1_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A71</font></span>: 
<font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</font></span></a></span>;<br/><a NAME="E7:101_1_1_6_1_2_1_2"/>
<font color="Maroon" title="c9">E1</font> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:101_1_1"><span class="lab"><font color="Green" title="E73">A12</font></span></a>, <a class="ref" href="xboole_0.html#D7" onmouseover="rs('xboole_0/D7')" onmouseout="rh()">XBOOLE_0:def 7</a></span>;<br/><a NAME="E8:101_1_1_6_1_2_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c14">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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E81">A69</font></span></a>, <a class="txt" href="glib_007.html#E35"><span class="lab"><font color="Green" title="E30">Lm7</font></span></a></span>;<br/><a NAME="E9:101_1_1_6_1_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E84">A72</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E83">A71</font></span></a>, <a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">e3</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E11:101_1_1_6_1_2_1_2"/>
( <font color="Maroon" title="c15">e3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">E2</font> &amp; <font color="Maroon" title="c15">e3</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font> )
 <span class="kw">and </span><br/><a NAME="E12:101_1_1_6_1_2_1_2"/><span class="lab"><font color="Green" title="E85">A73</font></span>: 
 for <font color="Olive" title="b1">e2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e2</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</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="c5">G2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">e</font>,<font color="Maroon" title="c2">v</font>,<font color="Maroon" title="c5">G2</font> holds <br/><font color="Maroon" title="c15">e3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:101_1_1"><span class="lab"><font color="Green" title="E76">A15</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E80">A68</font></span></a></span>;<br/><a NAME="E13:101_1_1_6_1_2_1_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">e3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E84">A72</font></span></a>, <a class="txt" href="#E12:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E85">A73</font></span></a></span>;<br/><a NAME="E14:101_1_1_6_1_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">e1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E85">A73</font></span></a>, <a class="txt" href="#E1:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E80">A68</font></span></a>, <a class="txt" href="#E2:101_1_1_6_1_2"><span class="lab"><font color="Green" title="E79">A60</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E15:101_1_1_6_1_2_1_2"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:101_1_1_6_1_2_1_2"><span class="lab"><font color="Green" title="E82">A70</font></span></a>, <a class="txt" href="#E7:101_1_1"><span class="lab"><font color="Green" title="E70">A9</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</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></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div><a NAME="E24:101_1_1"/><span class="kw">then </span>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E25:101_1_1"/>
<font color="Maroon" title="c8">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D3" onmouseover="rs('funct_2/D3')" onmouseout="rh()">FUNCT_2:def 3</a>, <a class="ref" href="xboole_0.html#D10" onmouseover="rs('xboole_0/D10')" onmouseout="rh()">XBOOLE_0:def 10</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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c12">e</font>, <font color="Maroon" title="c13">v2</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="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c4">G1</font> implies <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/><span class="kw">assume </span><a NAME="E26:101_1_1"/><span class="lab"><font color="Green" title="E77">A75</font></span>: 
<font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</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="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_7"><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="E1:101_1_1_7"/>
( <font color="Maroon" title="c12">e</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">G</font> or  not <font color="Maroon" title="c12">e</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">G</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_1_7_1"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_7_1"/><span class="lab"><font color="Green" title="E78">A76</font></span>: 
<font color="Maroon" title="c12">e</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">G</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="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><div class="add"><a NAME="E2:101_1_1_7_1"/><span class="kw">then </span>
<font color="Maroon" title="c12">e</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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>
 ;<br/><span class="kw">then </span><span class="lab"><font color="Green" title="E79">A77</font></span>: <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> = 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T13" onmouseover="rs('funct_4/T13')" onmouseout="rh()">FUNCT_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c12">e</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:101_1_1_7_1"><span class="lab"><font color="Green" title="E78">A76</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>
;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">w1</font> = <font color="Maroon" title="c11">v1</font>, <font color="Maroon" title="c15">w2</font> = <font color="Maroon" title="c13">v2</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="E5:101_1_1_7_1"/>
<font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c14">w1</font>,<font color="Maroon" title="c15">w2</font>,<font color="Maroon" title="c1">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:101_1_1"><span class="lab"><font color="Green" title="E77">A75</font></span></a>, <a class="txt" href="#E1:101_1_1_7_1"><span class="lab"><font color="Green" title="E78">A76</font></span></a>, <a class="ref" href="glib_006.html#T72" onmouseover="rs('glib_006/T72')" onmouseout="rh()">GLIB_006:72</a></span>;<br/><span class="kw">hence </span><a NAME="E6:101_1_1_7_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_7_1"><span class="lab"><font color="Green" title="E79">A77</font></span></a>, <a class="ref" href="glib_006.html#T70" onmouseover="rs('glib_006/T70')" onmouseout="rh()">GLIB_006:70</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="101_1_1_7_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_1_7_2"/>
 not <font color="Maroon" title="c12">e</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">G</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="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><div class="add"><a NAME="E2:101_1_1_7_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c12">e</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="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span></span>)</span>
 ;<br/><a NAME="E3:101_1_1_7_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E78">A78</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T11" onmouseover="rs('funct_4/T11')" onmouseout="rh()">FUNCT_4:11</a></span>;<br/><a NAME="E4:101_1_1_7_2"/>
<font color="Maroon" title="c12">e</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:101_1_1"><span class="lab"><font color="Green" title="E77">A75</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E5:101_1_1_7_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c11">v1</font>,<font color="Maroon" title="c13">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_1_7_2"><span class="lab"><font color="Green" title="E78">A78</font></span></a>, <a class="txt" href="#E26:101_1_1"><span class="lab"><font color="Green" title="E77">A75</font></span></a>, <a class="txt" href="#E4:101_1_1"><span class="lab"><font color="Green" title="E67">A4</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></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="101_1_2"><span class="kw">suppose </span></a><a NAME="E1:101_1_2"/>
(  not <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">G</font> or <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">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span> st <br/>( <font color="Olive" title="b1">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Olive" title="b1">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b2">v1</font>, <font color="Olive" title="b3">e</font>, <font color="Olive" title="b4">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Olive" title="b1">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b2">v1</font>,<font color="Olive" title="b4">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><div class="add"><a NAME="E2:101_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A79</font></span>: 
( <font color="Maroon" title="c4">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon" title="c5">G2</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c1">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_007.html#D4" target="_self" onmouseover="rs('glib_007/D4')" onmouseout="rh()">Def4</a></span>;<br/><a NAME="E3:101_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A80</font></span>: 
(  <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>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G2</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">G</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D34" onmouseover="rs('glib_000/D34')" onmouseout="rh()">GLIB_000:def 34</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c6">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">f</font> =  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <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="c4">G1</font></span>)</span>,<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="c5">G2</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2"><span class="lab"><font color="Green" title="E66">A80</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">f</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></span>)</span> &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><span class="kw">thus </span><a NAME="E5:101_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</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">G</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</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">G</font></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"> ( <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  &amp; (  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font> ) )</span><br/><span class="kw">thus </span><a NAME="E6:101_1_2"/>
( <font color="Maroon" title="c7">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp; <font color="Maroon" title="c7">f</font> is  <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2"><span class="lab"><font color="Green" title="E66">A80</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">  for <font color="Olive" title="b1">v1</font>, <font color="Olive" title="b2">e</font>, <font color="Olive" title="b3">v2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c4">G1</font> holds <br/><font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v1</font>,<font color="Olive" title="b3">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><span class="kw">let </span><font color="Maroon" title="c8">v1</font>, <font color="Maroon" title="c9">e</font>, <font color="Maroon" title="c10">v2</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">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c4">G1</font> implies <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c5">G2</font> )</span><br/><span class="kw">assume </span><a NAME="E7:101_1_2"/><span class="lab"><font color="Green" title="E67">A81</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</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="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c5">G2</font></span><br/><a NAME="E8:101_1_2"/>
<font color="Maroon" title="c4">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:101_1_2"><span class="lab"><font color="Green" title="E65">A79</font></span></a>, <a class="ref" href="glib_000.html#T85" onmouseover="rs('glib_000/T85')" onmouseout="rh()">GLIB_000:85</a></span>;<br/><a NAME="E9:101_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A82</font></span>: 
<font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2"><span class="lab"><font color="Green" title="E67">A81</font></span></a>, <a class="ref" href="glib_000.html#T88" onmouseover="rs('glib_000/T88')" onmouseout="rh()">GLIB_000:88</a></span>;<br/><a NAME="E10:101_1_2"/>
<font color="Maroon" title="c9">e</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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:101_1_2"><span class="lab"><font color="Green" title="E67">A81</font></span></a>, <a class="ref" href="glib_000.html#D13" onmouseover="rs('glib_000/D13')" onmouseout="rh()">GLIB_000:def 13</a></span>;<br/><span class="kw">hence </span><a NAME="E11:101_1_2"/>
<font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c8">v1</font>,<font color="Maroon" title="c10">v2</font>,<font color="Maroon" title="c5">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:101_1_2"><span class="lab"><font color="Green" title="E66">A80</font></span></a>, <a class="txt" href="#E9:101_1_2"><span class="lab"><font color="Green" title="E68">A82</font></span></a>, <a class="ref" href="funct_1.html#T18" onmouseover="rs('funct_1/T18')" onmouseout="rh()">FUNCT_1:18</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

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