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<span class="kw">let </span><font color="Maroon" title="c1">GS</font> be   <a href="glib_000.html#NM1" title="GLIB_000:NM.1">GraphStruct</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">GS</font> is  <a href="glib_000.html#V1" title="GLIB_000:attr.1">[Graph-like]</a>  iff (  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp; not  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="glib_000.html#K8" title="GLIB_000:func.8">the_Source_of</a> <font color="Maroon" title="c1">GS</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">GS</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font></span>)</span> &amp;  <a href="glib_000.html#K9" title="GLIB_000:func.9">the_Target_of</a> <font color="Maroon" title="c1">GS</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">GS</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font></span>)</span> ) )</span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><span class="kw">now </span></a><div class="add"><div><span class="kw">hereby </span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> implies (  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> ) )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:14_1_1"/>
(  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font></span><br/><a NAME="E2:14_1_1"/><span class="kw">then </span>
for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  holds <br/><font color="Olive" title="b1">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E3:14_1_1"/>
 <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div>
<span class="kw">end;</span></div><a NAME="E2:14_1"/><span class="lab"><font color="Green" title="E3">A1</font></span>: 
(  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  &amp;  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><a NAME="E3:14_1"/><span class="lab"><font color="Green" title="E4">A2</font></span>: 
(  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  &amp;  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#D2" title="ENUMSET1:def.2">ENUMSET1:def 2</a></span>;<br/><span class="kw">assume </span><a NAME="E4:14_1"/>
 <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> )</span><br/><span class="kw">hence </span><a NAME="E5:14_1"/>
(  <a href="glib_000.html#K1" title="GLIB_000:func.1">VertexSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K2" title="GLIB_000:func.2">EdgeSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K3" title="GLIB_000:func.3">SourceSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp;  <a href="glib_000.html#K4" title="GLIB_000:func.4">TargetSelector</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:14_1"><span class="lab"><font color="Green" title="E3">A1</font></span></a>, <a class="txt" href="#E3:14_1"><span class="lab"><font color="Green" title="E4">A2</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E2:14"/>
( <font color="Maroon" title="c1">GS</font> is  <a href="glib_000.html#V1" title="GLIB_000:attr.1">[Graph-like]</a>  iff (  <a href="glib_000.html#K5" title="GLIB_000:func.5">_GraphSelectors</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c1">GS</font> &amp; not  <a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font> is  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  &amp;  <a href="glib_000.html#K8" title="GLIB_000:func.8">the_Source_of</a> <font color="Maroon" title="c1">GS</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">GS</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font></span>)</span> &amp;  <a href="glib_000.html#K9" title="GLIB_000:func.9">the_Target_of</a> <font color="Maroon" title="c1">GS</font> is   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">GS</font></span>)</span>,<span class="p1">(<span class="default"><a href="glib_000.html#K6" title="GLIB_000:func.6">the_Vertices_of</a> <font color="Maroon" title="c1">GS</font></span>)</span> ) )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_000.html#D11" target="_self" title="GLIB_000:def.11">Def11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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