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<b>let </b><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" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Olive" title="b1">x</font> holds <br/><font color="Maroon" title="c2">G2</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Olive" title="b1">x</font></span><br/><b>let </b><font color="Maroon" title="c3">x</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="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Maroon" title="c3">x</font> implies <font color="Maroon" title="c2">G2</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Maroon" title="c3">x</font> )</span><br/>





<b>assume </b><a NAME="E1:82"/><i><font color="Green" title="E48">A1</font></i>: 
( <font color="Maroon" title="c1">G1</font> <a href="glib_000.html#R5" title="GLIB_000:pred.5">==</a> <font color="Maroon" title="c2">G2</font> &amp; <font color="Maroon" title="c1">G1</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Maroon" title="c3">x</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">G2</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Maroon" title="c3">x</font></span><br/>

<a NAME="E2:82"/><b>then </b>
( <font color="Maroon" title="c1">G1</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a> &amp; ( for <font color="Olive" title="b1">y</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">W</font> being  <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G1</font> st <font color="Olive" title="b2">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Olive" title="b1">y</font> ) )
 
<b>by </b><i><a class="ref" href="glib_002.html#D4" target="_self" title="GLIB_002:def.4">Def4</a></i>;<br/>
<a NAME="E3:82"/><b>then </b><i><font color="Green" title="E49">A2</font></i>: 
<font color="Maroon" title="c2">G2</font> is <a href="glib_002.html#V3" title="GLIB_002:attr.3">Tree-like</a>
 
<b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_002.html#T48" target="_self" title="GLIB_002:th.48">Th48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="82_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">y</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c2">G2</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">W'</font> being  <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c2">G2</font> st <font color="Olive" title="b1">W'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/><b>reconsider </b><font color="Maroon" title="c5">y'</font> = <font color="Maroon" title="c4">y</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">G1</font> <b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_000.html#D36" title="GLIB_000:def.36">GLIB_000:def 36</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">W</font> being   <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c1">G1</font><b> such that </b><br/><a NAME="E3:82_1"/><i><font color="Green" title="E50">A3</font></i>: 
<font color="Maroon" title="c6">W</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c5">y'</font>
 <b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_002.html#D4" target="_self" title="GLIB_002:def.4">Def4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c7">W'</font> = <font color="Maroon" title="c6">W</font> as   <a href="glib_001.html#NM6" title="GLIB_001:NM.6">DWalk</a> of <font color="Maroon" title="c2">G2</font> <b>by </b><i><a class="txt" href="#E1:82"><i><font color="Green" title="E48">A1</font></i></a>, <a class="ref" href="glib_001.html#T180" title="GLIB_001:th.180">GLIB_001:180</a>, <a class="ref" href="glib_001.html#T182" title="GLIB_001:th.182">GLIB_001:182</a></i>;<br/><b>take </b><font color="Maroon" title="c8">W'</font> = <font color="Maroon" title="c7">W'</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">W'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><br/><b>thus </b><a NAME="E5:82_1"/>
<font color="Maroon" title="c8">W'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font>
 <b>by </b><i><a class="txt" href="#E3:82_1"><i><font color="Green" title="E50">A3</font></i></a>, <a class="ref" href="glib_001.html#T20" title="GLIB_001:th.20">GLIB_001:20</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:82"/>
<font color="Maroon" title="c2">G2</font> <a href="glib_002.html#R1" title="GLIB_002:pred.1">is_DTree_rooted_at</a> <font color="Maroon" title="c3">x</font>
 <b>by </b><i><a class="txt" href="#E3:82"><i><font color="Green" title="E49">A2</font></i></a>, <a class="ref" href="glib_002.html#D4" target="_self" title="GLIB_002:def.4">Def4</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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