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<span class="kw">let </span><font color="Maroon" title="c1">G1</font>, <font color="Maroon" title="c2">G2</font> be   <a href="glib_000.html#NM2" title="GLIB_000:NM.2">_Graph</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">F0</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>  st <font color="Olive" title="b1">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  holds <br/> ex <font color="Olive" title="b2">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b3">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b2">E</font>  ex <font color="Olive" title="b4">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b3">G3</font> st <br/>( <font color="Olive" title="b4">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F0</font> &amp; <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Olive" title="b1">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b4">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">F0</font> be    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c2">G2</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  implies  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:110"/><span class="lab"><font color="Green" title="E29">A1</font></span>: 
<font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a>  is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</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">  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_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="E1:110_1"/>
(  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">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>  or  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">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>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">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> 
 ; <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">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><div class="add"><span class="kw">set </span><font color="Maroon" title="c4">E</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e9</font> where <font color="Olive" title="b1">e9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> : ( <font color="Olive" title="b1">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; (  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> holds <br/><font color="Olive" title="b1">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font> ) ) </span> } </span> ;<br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b2">e9</font> where <font color="Olive" title="b2">e9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> : ( <font color="Olive" title="b2">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; (  for <font color="Olive" title="b3">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b3">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">e9</font> holds <br/><font color="Olive" title="b2">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font> ) ) </span> } </span>  holds <br/><font color="Olive" title="b1">x</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="c2">G2</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e9</font> where <font color="Olive" title="b1">e9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> : ( <font color="Olive" title="b1">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; (  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> holds <br/><font color="Olive" title="b1">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font> ) ) </span> } </span>  implies <font color="Maroon" title="c5">x</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="c2">G2</font> )</span><br/><span class="kw">assume </span><a NAME="E1:110_1_1_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e9</font> where <font color="Olive" title="b1">e9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> : ( <font color="Olive" title="b1">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; (  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> holds <br/><font color="Olive" title="b1">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<font color="Maroon" title="c2">G2</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="c5">x</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="c2">G2</font></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">e9</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E3:110_1_1_1"/><span class="lab"><font color="Green" title="E30">A2</font></span>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e9</font>
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_1"/><span class="lab"><font color="Green" title="E31">A3</font></span>: 
<font color="Maroon" title="c6">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:110_1_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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">e9</font> holds <br/><font color="Maroon" title="c6">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font>
 ;<br/><a NAME="E6:110_1_1_1"/>
<font color="Maroon" title="c6">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_1"><span class="lab"><font color="Green" title="E31">A3</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:110_1_1_1"/>
<font color="Maroon" title="c5">x</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="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_1"><span class="lab"><font color="Green" title="E30">A2</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">then </span><span class="kw">reconsider </span><font color="Maroon" title="c5">E</font> = <span class="p1"> { <span class="default"> <font color="Olive" title="b1">e9</font> where <font color="Olive" title="b1">e9</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> : ( <font color="Olive" title="b1">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; (  for <font color="Olive" title="b2">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b2">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">e9</font> holds <br/><font color="Olive" title="b1">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font> ) ) </span> } </span>  as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> <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">take </span>
<font color="Maroon" title="c5">E</font>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">E</font>  ex <font color="Olive" title="b2">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c6">G3</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">E</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="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c6">G3</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c7">F9</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c6">G3</font><span class="kw"> such that </span><br/><a NAME="E5:110_1_1"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
( <font color="Maroon" title="c7">F9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c7">F9</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T77" target="_self" onmouseover="rs('glibpre0/T77')" onmouseout="rh()">Th83</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</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">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><span class="lab"><font color="Green" title="E31">A5</font></span>: <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> = 
<span class="p1">(<span class="default"><a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">F0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#T116" onmouseover="rs('glib_010/T116')" onmouseout="rh()">GLIB_010:116</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="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">e</font>, <font color="Olive" title="b2">v</font>, <font color="Olive" title="b3">w</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="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Olive" title="b3">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Olive" title="b2">v</font>,<font color="Olive" title="b3">w</font>,<font color="Maroon" title="c1">G1</font> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">w</font>,<font color="Maroon" title="c6">G3</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c8">e</font>, <font color="Maroon" title="c9">v</font>, <font color="Maroon" title="c10">w</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="c8">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G1</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c6">G3</font> )</span><br/><span class="kw">assume </span><a NAME="E1:110_1_1_3"/>
( <font color="Maroon" title="c8">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"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></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">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G1</font> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c6">G3</font> )</span><br/><a NAME="E2:110_1_1_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></span>: 
( <font color="Maroon" title="c8">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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Maroon" title="c9">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> &amp; <font color="Maroon" title="c10">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E31">A5</font></span></a></span>;<br/><span class="kw">assume </span><a NAME="E3:110_1_1_3"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c6">G3</font></span><br/><a NAME="E4:110_1_1_3"/><span class="lab"><font color="Green" title="E34">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c11">e9</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font> ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_3_1"><span class="kw">per </span></a><a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:110_1_1_3_1"/>
( <font color="Maroon" title="c11">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">E</font> or  not <font color="Maroon" title="c11">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">E</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="110_1_1_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_3_1_1"/><span class="lab"><font color="Green" title="E35">A11</font></span>: 
<font color="Maroon" title="c11">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">E</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c6">G3</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">e8</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">G2</font><span class="kw"> such that </span><br/><a NAME="E3:110_1_1_3_1_1"/><span class="lab"><font color="Green" title="E36">A12</font></span>: 
( <font color="Maroon" title="c12">e8</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e9</font> &amp; <font color="Maroon" title="c12">e8</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E4:110_1_1_3_1_1"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
 for <font color="Olive" title="b1">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e0</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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e8</font> holds <br/><font color="Maroon" title="c12">e8</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font></span>)</span>,<font color="Maroon" title="c2">G2</font>
 ;<br/><a NAME="E5:110_1_1_3_1_1"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
<font color="Maroon" title="c11">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font></span>)</span>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E3:110_1_1_3_1_1"><span class="lab"><font color="Green" title="E36">A12</font></span></a>, <a class="txt" href="#E4:110_1_1_3_1_1"><span class="lab"><font color="Green" title="E37">A13</font></span></a></span>;<br/><a NAME="E6:110_1_1_3_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="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_3"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/><a NAME="E7:110_1_1_3_1_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_3_1_1"><span class="lab"><font color="Green" title="E35">A11</font></span></a>, <a class="txt" href="#E5:110_1_1_3_1_1"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="glib_007.html#T7" onmouseover="rs('glib_007/T7')" onmouseout="rh()">GLIB_007:7</a></span>;<br/><span class="kw">hence </span><a NAME="E8:110_1_1_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E31">A5</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="110_1_1_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_1_3_1_2"/><span class="lab"><font color="Green" title="E35">A15</font></span>: 
 not <font color="Maroon" title="c11">e9</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c5">E</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"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>2</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon" title="c6">G3</font></span><br/><div class="add"><a NAME="E2:110_1_1_3_1_2"/>
<font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_3"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E3:110_1_1_3_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">A16</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="glib_010.html#T4" onmouseover="rs('glib_010/T4')" onmouseout="rh()">GLIB_010:4</a></span>;<br/><a NAME="E4:110_1_1_3_1_2"/>
 not <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c2">G2</font>
 <div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_3_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:110_1_1_3_1_2_1"/><span class="lab"><font color="Green" title="E37">A17</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<font color="Maroon" title="c2">G2</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:110_1_1_3_1_2_1"/>
 for <font color="Olive" title="b1">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>   st <font color="Olive" title="b1">e0</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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e9</font> holds <br/><font color="Maroon" title="c11">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">e0</font></span>)</span>,<font color="Maroon" title="c2">G2</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="110_1_1_3_1_2_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c12">e0</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">e0</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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e9</font> implies <font color="Maroon" title="c11">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<font color="Maroon" title="c2">G2</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:110_1_1_3_1_2_1_1"/>
( <font color="Maroon" title="c12">e0</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"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e9</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">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<font color="Maroon" title="c2">G2</font></span><br/>

<a NAME="E2:110_1_1_3_1_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">e0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110"><span class="lab"><font color="Green" title="E29">A1</font></span></a>, <a class="txt" href="#E2:110_1_1_3"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="funct_1.html#D4" onmouseover="rs('funct_1/D4')" onmouseout="rh()">FUNCT_1:def 4</a></span>;<br/>
<a NAME="E3:110_1_1_3_1_2_1_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">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="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_3"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="glib_000.html#D14" onmouseover="rs('glib_000/D14')" onmouseout="rh()">GLIB_000:def 14</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:110_1_1_3_1_2_1_1"/>
<font color="Maroon" title="c11">e9</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K11" title="GLIB_000:func.11">the_Target_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="glib_000.html#K10" title="GLIB_000:func.10">the_Source_of</a> <font color="Maroon" title="c1">G1</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">e0</font></span>)</span>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_3_1_2_1"><span class="lab"><font color="Green" title="E37">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>
<span class="kw">hence </span><a NAME="E3:110_1_1_3_1_2_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_1_3"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E1:110_1_1_3_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E5:110_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_1_3_1_2"><span class="lab"><font color="Green" title="E36">A16</font></span></a>, <a class="ref" href="glib_000.html#T16" onmouseover="rs('glib_000/T16')" onmouseout="rh()">GLIB_000:16</a></span>;<br/><a NAME="E6:110_1_1_3_1_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F0</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:110_1_1_3_1_2"><span class="lab"><font color="Green" title="E35">A15</font></span></a>, <a class="ref" href="glib_007.html#T8" onmouseover="rs('glib_007/T8')" onmouseout="rh()">GLIB_007:8</a></span>;<br/><span class="kw">hence </span><a NAME="E7:110_1_1_3_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">e</font> <a href="glib_000.html#R2" title="GLIB_000:pred.2">DJoins</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">v</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font></span>)</span> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c10">w</font>,<font color="Maroon" title="c6">G3</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E31">A5</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><span class="kw">hence </span><a NAME="E8:110_1_1"/>
<font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#D14" onmouseover="rs('glib_010/D14')" onmouseout="rh()">GLIB_010:def 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"> ( (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E9:110_1_1"/>
(  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:110_1_1"><span class="lab"><font color="Green" title="E31">A5</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"> ( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E10:110_1_1"/>
( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c7">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:110_1_1"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="ref" href="glib_010.html#T104" onmouseover="rs('glib_010/T104')" onmouseout="rh()">GLIB_010:104</a>, <a class="ref" href="glib_010.html#T106" onmouseover="rs('glib_010/T106')" onmouseout="rh()">GLIB_010:106</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="110_1_2"><span class="kw">suppose </span></a><a NAME="E1:110_1_2"/><span class="lab"><font color="Green" title="E30">A18</font></span>: 
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c2">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">  ex <font color="Olive" title="b1">E</font> being   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</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="c2">G2</font></span>)</span> st <br/> for <font color="Olive" title="b2">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Olive" title="b1">E</font>  ex <font color="Olive" title="b3">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b2">G3</font> st <br/>( <font color="Olive" title="b3">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b3">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c4">E</font> =  <a href="subset_1.html#K1" title="SUBSET_1:func.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="c2">G2</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">  for <font color="Olive" title="b1">G3</font> being    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">E</font>  ex <font color="Olive" title="b2">F</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Olive" title="b1">G3</font> st <br/>( <font color="Olive" title="b2">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b2">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c5">G3</font> be    <a href="glib_007.html#M1" title="GLIB_007:mode.1">reverseEdgeDirections</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c4">E</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="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c1">G1</font>,<font color="Maroon" title="c5">G3</font> st <br/>( <font color="Olive" title="b1">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">consider </span><font color="Maroon" title="c6">F9</font> being    <a href="glib_010.html#M1" title="GLIB_010:mode.1">PGraphMapping</a> of <font color="Maroon" title="c2">G2</font>,<font color="Maroon" title="c5">G3</font><span class="kw"> such that </span><br/><a NAME="E3:110_1_2"/><span class="lab"><font color="Green" title="E31">A19</font></span>: 
( <font color="Maroon" title="c6">F9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c6">F9</font> is  <a href="glib_010.html#V17" title="GLIB_010:attr.17">isomorphism</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glibpre0.html#T77" target="_self" onmouseover="rs('glibpre0/T77')" onmouseout="rh()">Th83</a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</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="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">F0</font> &amp; <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><span class="lab"><font color="Green" title="E32">A20</font></span>: <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> = 
<span class="p1">(<span class="default"><a href="glib_010.html#K7" title="GLIB_010:func.7">id</a> <font color="Maroon" title="c2">G2</font></span>)</span> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c3">F0</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="glib_010.html#T116" onmouseover="rs('glib_010/T116')" onmouseout="rh()">GLIB_010:116</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="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a>  &amp; (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><a NAME="E5:110_1_2"/>
 <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c5">G3</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="#E1:110_1_2"><span class="lab"><font color="Green" title="E30">A18</font></span></a>, <a class="ref" href="glib_007.html#T4" onmouseover="rs('glib_007/T4')" onmouseout="rh()">GLIB_007:4</a></span>;<br/><a NAME="E6:110_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c5">G3</font> is  <a href="glib_008.html#V1" title="GLIB_008:attr.1">edgeless</a> 
 ;<br/><span class="kw">hence </span><a NAME="E7:110_1_2"/>
<font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V10" title="GLIB_010:attr.10">directed</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E8:110_1_2"/>
(  not <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  implies  not <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V6" title="GLIB_010:attr.6">empty</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:110_1_2"><span class="lab"><font color="Green" title="E32">A20</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"> ( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )</span><br/><span class="kw">thus </span><a NAME="E9:110_1_2"/>
( ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V7" title="GLIB_010:attr.7">total</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V9" title="GLIB_010:attr.9">one-to-one</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V8" title="GLIB_010:attr.8">onto</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  ) &amp; ( <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  implies <font color="Maroon" title="c6">F9</font> <a href="glib_010.html#K13" title="GLIB_010:func.13">*</a> <font color="Maroon" title="c3">F0</font> is  <a href="glib_010.html#V12" title="GLIB_010:attr.12">continuous</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:110_1_2"><span class="lab"><font color="Green" title="E31">A19</font></span></a>, <a class="ref" href="glib_010.html#T104" onmouseover="rs('glib_010/T104')" onmouseout="rh()">GLIB_010:104</a>, <a class="ref" href="glib_010.html#T106" onmouseover="rs('glib_010/T106')" onmouseout="rh()">GLIB_010:106</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>
