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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">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="c2">G2</font>  st <font color="Olive" title="b1">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Maroon" title="c2">G2</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  holds <br/><font color="Maroon" title="c1">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c3">F</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">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> &amp; <font color="Maroon" title="c2">G2</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  implies <font color="Maroon" title="c1">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:12"/><span class="lab"><font color="Green" title="E9">A1</font></span>: 
<font color="Maroon" title="c3">F</font> is  <a href="glib_010.html#V11" title="GLIB_010:attr.11">semi-continuous</a> 
 <span class="kw">and </span><br/><a NAME="E2:12"/><span class="lab"><font color="Green" title="E10">A2</font></span>: 
(  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</font> <a href="glib_010.html#K5" title="GLIB_010:func.5">_V</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">G1</font> &amp; <font color="Maroon" title="c2">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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">F</font> <a href="glib_010.html#K6" title="GLIB_010:func.6">_E</a></span>)</span> )
 <span class="kw">and </span><br/><a NAME="E3:12"/><span class="lab"><font color="Green" title="E11">A3</font></span>: 
<font color="Maroon" title="c2">G2</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</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="c1">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> </span><br/>

<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="12_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">v</font> being   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>  ex <font color="Olive" title="b2">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a>  st <font color="Olive" title="b2">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Olive" title="b1">v</font>,<font color="Olive" title="b1">v</font>,<font color="Maroon" title="c1">G1</font></span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c4">v</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  ex <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="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G1</font></span><br/><a NAME="E1:12_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</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="c4">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">F</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="#E2:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_1.html#T3" onmouseover="rs('funct_1/T3')" onmouseout="rh()">FUNCT_1:3</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c5">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:12_1"/><span class="lab"><font color="Green" title="E12">A4</font></span>: 
<font color="Maroon" title="c5">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">F</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="c4">v</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">F</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="c4">v</font>,<font color="Maroon" title="c2">G2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12"><span class="lab"><font color="Green" title="E11">A3</font></span></a></span>;<br/><a NAME="E4:12_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">G2</font> <a href="glib_009.html#K1" title="GLIB_009:func.1">.loops()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="ref" href="glib_009.html#D2" onmouseover="rs('glib_009/D2')" onmouseout="rh()">GLIB_009:def 2</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">e0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E6:12_1"/><span class="lab"><font color="Green" title="E13">A5</font></span>: 
( <font color="Maroon" title="c6">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">F</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">F</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="c6">e0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">e</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="ref" href="funct_1.html#D3" onmouseover="rs('funct_1/D3')" onmouseout="rh()">FUNCT_1:def 3</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c7">e0</font> = <font color="Maroon" title="c6">e0</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">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G1</font></span><br/><span class="kw">thus </span><a NAME="E7:12_1"/>
<font color="Maroon" title="c7">e0</font> <a href="glib_000.html#R1" title="GLIB_000:pred.1">Joins</a> <font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c4">v</font>,<font color="Maroon" title="c1">G1</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:12"><span class="lab"><font color="Green" title="E9">A1</font></span></a>, <a class="txt" href="#E2:12"><span class="lab"><font color="Green" title="E10">A2</font></span></a>, <a class="txt" href="#E3:12_1"><span class="lab"><font color="Green" title="E12">A4</font></span></a>, <a class="txt" href="#E6:12_1"><span class="lab"><font color="Green" title="E13">A5</font></span></a>, <a class="ref" href="glib_010.html#D15" onmouseover="rs('glib_010/D15')" onmouseout="rh()">GLIB_010:def 15</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E5:12"/>
<font color="Maroon" title="c1">G1</font> is  <a href="glib_012.html#V1" title="GLIB_012:attr.1">loopfull</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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