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<b>let </b><font color="Maroon" title="c1">T</font> be   <a href="helly.html#NM1" title="HELLY:NM.1">_Tree</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">a</font>, <font color="Olive" title="b2">b</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b3">t</font> being  <a href="helly.html#NM2" title="HELLY:NM.2">_Subtree</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b4">a'</font>, <font color="Olive" title="b5">b'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b3">t</font>  st <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">a'</font> &amp; <font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">b'</font> holds <br/><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b4">a'</font>,<font color="Olive" title="b5">b'</font></span><br/><b>let </b><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</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">t</font> being  <a href="helly.html#NM2" title="HELLY:NM.2">_Subtree</a> of <font color="Maroon" title="c1">T</font><br/> for <font color="Olive" title="b2">a'</font>, <font color="Olive" title="b3">b'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Olive" title="b1">t</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a'</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">b'</font> holds <br/><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b2">a'</font>,<font color="Olive" title="b3">b'</font></span><br/><b>let </b><font color="Maroon" title="c4">t</font> be   <a href="helly.html#NM2" title="HELLY:NM.2">_Subtree</a> of <font color="Maroon" title="c1">T</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">a'</font>, <font color="Olive" title="b2">b'</font> being  <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">t</font>  st <font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">a'</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">b'</font> holds <br/><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b1">a'</font>,<font color="Olive" title="b2">b'</font></span><br/><b>let </b><font color="Maroon" title="c5">a'</font>, <font color="Maroon" title="c6">b'</font> be   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c4">t</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">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font> implies <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> )</span><br/>







<b>assume </b><b>that </b><br/><a NAME="E1:41"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c2">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">a'</font>
 <b>and </b><br/><a NAME="E2:41"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">b'</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span><br/>

<b>set </b><font color="Maroon" title="c7">tp</font> = <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>;<br/>
<a NAME="E3:41"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
 
<b>by </b><i><a class="ref" href="helly.html#D2" target="_self" title="HELLY:def.2">Def2</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">tp'</font> = <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font> as    <a href="glib_001.html#M3" title="GLIB_001:mode.3">Walk</a> of <font color="Maroon" title="c1">T</font> <b>by </b><i><a class="ref" href="glib_001.html#T168" title="GLIB_001:th.168">GLIB_001:168</a></i>;<br/>
<a NAME="E5:41"/><i><font color="Green" title="E34">A4</font></i>: 
<font color="Maroon" title="c8">tp'</font> is <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>
 
<b>by </b><i><a class="ref" href="glib_001.html#T177" title="GLIB_001:th.177">GLIB_001:177</a></i>;<br/>
<i><font color="Green" title="E35">A5</font></i>: <font color="Maroon" title="c8">tp'</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

<br/>.= 
<font color="Maroon" title="c2">a</font>
<b>by </b><i><a class="txt" href="#E1:41"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="#E3:41"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<font color="Maroon" title="c8">tp'</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  = 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font></span>)</span> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> 

<br/>.= 
<font color="Maroon" title="c3">b</font>
<b>by </b><i><a class="txt" href="#E2:41"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="#E3:41"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>
;<br/>
<a NAME="E8:41"/><b>then </b>
<font color="Maroon" title="c8">tp'</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font>
 
<b>by </b><i><a class="txt" href="#E6:41"><i><font color="Green" title="E35">A5</font></i></a>, <a class="ref" href="glib_001.html#D23" title="GLIB_001:def.23">GLIB_001:def 23</a></i>;<br/>
<b>hence </b><a NAME="E9:41"/>
<font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">t</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">a'</font>,<font color="Maroon" title="c6">b'</font>
 <b>by </b><i><a class="txt" href="#E5:41"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="helly.html#D2" target="_self" title="HELLY:def.2">Def2</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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