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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>( $1 <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> or $1 <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> );<br/>
<b>consider </b><font color="Maroon" title="c4">IT</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="c1">G</font></span>)</span><b> such that </b><br/><a NAME="E2:20_1_1"/><i><font color="Green" title="E16">A1</font></i>: 
for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">e</font>] ) )
 <b>from </b><i><a class="ref" href="subset_1.html#S1" title="SUBSET_1:sch.1">SUBSET_1:sch 1</a>();<br/></i>
<b>take </b>
<font color="Maroon" title="c4">IT</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">e</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  holds <br/> ( <font color="Olive" title="b1">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff ( <font color="Olive" title="b1">e</font> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> or <font color="Olive" title="b1">e</font> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> ) )</span><br/>

<b>let </b><font color="Maroon" title="c5">e</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> iff ( <font color="Maroon" title="c5">e</font> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> or <font color="Maroon" title="c5">e</font> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> ) )</span><br/>

<b>thus </b><a NAME="E3:20_1_1"/>
( <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">e</font>] )
 <b>by </b><i><a class="txt" href="#E2:20_1_1"><i><font color="Green" title="E16">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( ( <font color="Maroon" title="c5">e</font> <a href="glib_005.html#R3" title="GLIB_005:pred.3">is_forward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> or <font color="Maroon" title="c5">e</font> <a href="glib_005.html#R4" title="GLIB_005:pred.4">is_backward_edge_wrt</a> <font color="Maroon" title="c3">VL</font> ) implies <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font> )</span><br/>

<b>assume </b><a NAME="E4:20_1_1"/><i><font color="Green" title="E17">A2</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">e</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font></span><br/>

<a NAME="E5:20_1_1"/><b>then </b>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="glib_000.html#K7" title="GLIB_000:func.7">the_Edges_of</a> <font color="Maroon" title="c1">G</font>
 
<b>by </b><i><a class="ref" href="glib_005.html#D6" target="_self" title="GLIB_005:def.6">Def10</a>, <a class="ref" href="glib_005.html#D7" target="_self" title="GLIB_005:def.7">Def11</a></i>;<br/>
<b>hence </b><a NAME="E6:20_1_1"/>
<font color="Maroon" title="c5">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">IT</font>
 <b>by </b><i><a class="txt" href="#E2:20_1_1"><i><font color="Green" title="E16">A1</font></i></a>, <a class="txt" href="#E4:20_1_1"><i><font color="Green" title="E17">A2</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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