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<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being  <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> holds <br/><font color="Olive" title="b1">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Olive" title="b2">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</font> be   <a href="glib_001.html#NM5" title="GLIB_001:NM.5">Path</a> of <font color="Maroon" title="c1">T</font>; <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="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> implies <font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:46"/><span class="lab"><font color="Green" title="E36">A1</font></span>: 
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 
 <span class="kw">and </span><br/><a NAME="E2:46"/><span class="lab"><font color="Green" title="E37">A2</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K4" title="GLIB_001:func.4">.last()</a> </span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 ; <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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:46_1"/>
( <font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  or ( not <font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  &amp; not <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_1"><span class="kw">suppose </span></a><a NAME="E1:46_1_1"/>
<font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_1"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="ref" href="helly.html#T16" target="_self" title="HELLY:th.16">Th16</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_2"><span class="kw">suppose </span></a><a NAME="E1:46_1_2"/>
<font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_2"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="glib_001.html#T131" title="GLIB_001:th.131">GLIB_001:131</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="46_1_3"><span class="kw">suppose </span></a><a NAME="E1:46_1_3"/>
( not <font color="Maroon" title="c2">P1</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</a>  &amp; not <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V3" title="GLIB_001:attr.3">trivial</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="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> </span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:46_1_3"/>
<font color="Maroon" title="c2">P1</font> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <font color="Maroon" title="c3">P2</font> is  <a href="glib_001.html#V5" title="GLIB_001:attr.5">Path-like</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:46"><span class="lab"><font color="Green" title="E36">A1</font></span></a>, <a class="txt" href="#E2:46"><span class="lab"><font color="Green" title="E37">A2</font></span></a>, <a class="ref" href="helly.html#T19" target="_self" title="HELLY:th.19">Th19</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></div><span class="kw">end;</span></div>

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