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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">a</font>, <font color="Olive" title="b2">b</font>, <font color="Olive" title="b3">c</font>, <font color="Olive" title="b4">d</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="b5">P1</font>, <font color="Olive" title="b6">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="b5">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Olive" title="b6">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="b3">c</font> &amp; not <font color="Olive" title="b5">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b6">P2</font> &amp; not <font color="Olive" title="b6">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b5">P1</font> &amp; <font color="Olive" title="b4">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b5">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Olive" title="b5">P1</font>,<font color="Olive" title="b6">P2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b2">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Olive" title="b4">d</font>,<font color="Olive" title="b3">c</font></span>)</span> <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"><font color="Olive" title="b4">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <font color="Maroon" title="c3">b</font>, <font color="Maroon" title="c4">c</font>, <font color="Maroon" title="c5">d</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:()"><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="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Olive" title="b2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">c</font> &amp; not <font color="Olive" title="b1">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b2">P2</font> &amp; not <font color="Olive" title="b2">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">P1</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Olive" title="b1">P1</font>,<font color="Olive" title="b2">P2</font></span>)</span></span>)</span> holds <br/><span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/><span class="kw">let </span><font color="Maroon" title="c6">P1</font>, <font color="Maroon" title="c7">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="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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> &amp; <font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">c</font> &amp; not <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c7">P2</font> &amp; not <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Maroon" title="c6">P1</font> &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c7">P2</font></span>)</span></span>)</span> implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:48"/><span class="lab"><font color="Green" title="E38">A1</font></span>: 
<font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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>
 <span class="kw">and </span><br/><a NAME="E2:48"/><span class="lab"><font color="Green" title="E39">A2</font></span>: 
<font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <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="c4">c</font>
 <span class="kw">and </span><br/><a NAME="E3:48"/><span class="lab"><font color="Green" title="E40">A3</font></span>: 
not <font color="Maroon" title="c6">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P2</font>
 <span class="kw">and </span><br/><a NAME="E4:48"/><span class="lab"><font color="Green" title="E41">A4</font></span>: 
not <font color="Maroon" title="c7">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c6">P1</font>
 <span class="kw">and </span><br/><a NAME="E5:48"/><span class="lab"><font color="Green" title="E42">A5</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c7">P2</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<span class="kw">set </span><font color="Maroon" title="c8">Pad</font> = <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="c5">d</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">di</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c7">P2</font></span>)</span>;<br/>
<a NAME="E6:48"/><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c6">P1</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>;<br/>
<a NAME="E7:48"/><span class="lab"><font color="Green" title="E44">A7</font></span>: 
<font color="Maroon" title="c7">P2</font> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">di</font> =  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c7">P2</font></span>)</span> as   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="ref" href="helly.html#T22" target="_self" title="HELLY:th.22">Th22</a></span>;<br/>
<a NAME="E9:48"/><span class="lab"><font color="Green" title="E45">A8</font></span>: 
<font color="Maroon" title="c10">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c11">Pdb</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font>;<br/>
<a NAME="E10:48"/><span class="lab"><font color="Green" title="E46">A9</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">Pdc</font> = <font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font>;<br/>
<a NAME="E11:48"/><span class="lab"><font color="Green" title="E47">A10</font></span>: 
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><a href="helly.html#K1" title="HELLY:func.1">maxPrefix</a> <font color="Maroon" title="c6">P1</font>,<font color="Maroon" title="c7">P2</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="helly.html#T7" target="_self" title="HELLY:th.7">Th7</a></span>;<br/>
<a NAME="E12:48"/><span class="lab"><font color="Green" title="E48">A11</font></span>: 
<font color="Maroon" title="c10">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T11" title="NAT_1:th.11">NAT_1:11</a></span>;<br/>
<a NAME="E13:48"/>
<font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="helly.html#T23" target="_self" title="HELLY:th.23">Th23</a></span>;<br/>
<a NAME="E14:48"/><span class="kw">then </span>
<font color="Maroon" title="c10">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">P2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48"><span class="lab"><font color="Green" title="E48">A11</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E15:48"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P2</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:48"><span class="lab"><font color="Green" title="E47">A10</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E16:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A12</font></span>: 
<font color="Maroon" title="c7">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c5">d</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:48"><span class="lab"><font color="Green" title="E39">A2</font></span></a>, <a class="ref" href="helly.html#T36" target="_self" title="HELLY:th.36">Th36</a></span>;<br/>
<a NAME="E17:48"/><span class="lab"><font color="Green" title="E50">A13</font></span>: 
<span class="p1">(<span class="default"><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="c5">d</font></span>)</span> <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="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>;<br/>
<span class="lab"><font color="Green" title="E51">A14</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

<br/>.= 
<font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>
;<br/>
<a NAME="E19:48"/><span class="lab"><font color="Green" title="E52">A15</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>;<br/>
<a NAME="E20:48"/>
<font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="ref" href="helly.html#T23" target="_self" title="HELLY:th.23">Th23</a></span>;<br/>
<a NAME="E21:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A16</font></span>: 
<font color="Maroon" title="c10">di</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">P1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48"><span class="lab"><font color="Green" title="E45">A8</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/>
<a NAME="E22:48"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">P1</font> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/>
<a NAME="E23:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E54">A17</font></span>: 
<font color="Maroon" title="c6">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><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="c5">d</font></span>)</span> <a href="glib_001.html#K7" title="GLIB_001:func.7">.append</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48"><span class="lab"><font color="Green" title="E38">A1</font></span></a>, <a class="ref" href="helly.html#T36" target="_self" title="HELLY:th.36">Th36</a></span>;<br/>
<a NAME="E24:48"/><span class="lab"><font color="Green" title="E55">A18</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">di</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/>
<a NAME="E25:48"/><span class="kw">then </span>
<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="c5">d</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">P1</font> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>,<font color="Maroon" title="c10">di</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48"><span class="lab"><font color="Green" title="E42">A5</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="ref" href="helly.html#T35" target="_self" title="HELLY:th.35">Th35</a></span>;<br/>
<a NAME="E26:48"/><span class="kw">then </span><span class="lab"><font color="Green" title="E56">A19</font></span>: 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p2">(<span class="default"><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="c5">d</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:48"><span class="lab"><font color="Green" title="E53">A16</font></span></a>, <a class="txt" href="#E24:48"><span class="lab"><font color="Green" title="E55">A18</font></span></a>, <a class="ref" href="glib_001.html#T37" title="GLIB_001:th.37">GLIB_001:37</a></span>;<br/>
<span class="lab"><font color="Green" title="E57">A20</font></span>: <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K3" title="GLIB_001:func.3">.first()</a> 

<br/>.= 
<font color="Maroon" title="c5">d</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T28" target="_self" title="HELLY:th.28">Th28</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E28:48"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <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"><font color="Maroon" title="c5">d</font></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"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3"><span class="kw">proof </span></a><div class="add">

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span></span>
<div class="add"><span class="kw">assume </span><a NAME="E1:48_3_1"/>
not <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">d</font></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"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">e</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E3:48_3_1"/><span class="lab"><font color="Green" title="E58">A21</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:48_3_1"/><span class="lab"><font color="Green" title="E59">A22</font></span>: 
not <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><a NAME="E5:48_3_1"/><span class="lab"><font color="Green" title="E60">A23</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_3_1"><span class="lab"><font color="Green" title="E58">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E6:48_3_1"/><span class="lab"><font color="Green" title="E61">A24</font></span>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_3_1"><span class="lab"><font color="Green" title="E58">A21</font></span></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c14">e</font> = <font color="Maroon" title="c13">e</font> as   <a href="glib_000.html#NM10" title="GLIB_000:NM.10">Vertex</a> of <font color="Maroon" title="c1">T</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48_3_1"><span class="lab"><font color="Green" title="E58">A21</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c15">ebi</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E9:48_3_1"/><span class="lab"><font color="Green" title="E62">A25</font></span>: 
<font color="Maroon" title="c15">ebi</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E10:48_3_1"/><span class="lab"><font color="Green" title="E63">A26</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">ebi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:48_3_1"><span class="lab"><font color="Green" title="E60">A23</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">Pdeb</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font>;<br/><a NAME="E11:48_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">ebi</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E12:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A27</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E27:48"><span class="lab"><font color="Green" title="E57">A20</font></span></a>, <a class="txt" href="#E9:48_3_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="txt" href="#E10:48_3_1"><span class="lab"><font color="Green" title="E63">A26</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E13:48_3_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:48_3_1_1"/><span class="lab"><font color="Green" title="E65">A28</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1_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="E1:48_3_1_1_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_3_1_1"><span class="lab"><font color="Green" title="E65">A28</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_3_1_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_3_1_1_1_1"/>
contradiction
 ; <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="48_3_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_3_1_1_1_2"/><span class="lab"><font color="Green" title="E66">A29</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_3_1_1_1_2"/><span class="lab"><font color="Green" title="E67">A30</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_3_1"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E3:48_3_1_1_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_3_1"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E1:48_3_1_1_1_2"><span class="lab"><font color="Green" title="E66">A29</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_3_1_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_3_1"><span class="lab"><font color="Green" title="E59">A22</font></span></a>, <a class="txt" href="#E2:48_3_1_1_1_2"><span class="lab"><font color="Green" title="E67">A30</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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>

</div><span class="kw">end;</span></div><a NAME="E14:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A31</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" title="CHORD:th.4">CHORD:4</a></span>;<br/><a NAME="E15:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A32</font></span>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c17">eci</font> being   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <span class="kw"> such that </span><br/><a NAME="E17:48_3_1"/><span class="lab"><font color="Green" title="E67">A33</font></span>: 
<font color="Maroon" title="c17">eci</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 <span class="kw">and </span><br/><a NAME="E18:48_3_1"/><span class="lab"><font color="Green" title="E68">A34</font></span>: 
<font color="Maroon" title="c14">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c17">eci</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:48_3_1"><span class="lab"><font color="Green" title="E61">A24</font></span></a>, <a class="ref" href="glib_001.html#T88" title="GLIB_001:th.88">GLIB_001:88</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c18">Pdec</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font>;<br/><a NAME="E19:48_3_1"/>
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c17">eci</font> &amp; <span class="p1">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 is   <a href="abian.html#NV2" title="ABIAN:NV.2">odd</a>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="abian.html#T12" title="ABIAN:th.12">ABIAN:12</a></span>;<br/><a NAME="E20:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A35</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font> <a href="glib_001.html#R1" title="GLIB_001:pred.1">is_Walk_from</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E18:48"><span class="lab"><font color="Green" title="E51">A14</font></span></a>, <a class="txt" href="#E17:48_3_1"><span class="lab"><font color="Green" title="E67">A33</font></span></a>, <a class="txt" href="#E18:48_3_1"><span class="lab"><font color="Green" title="E68">A34</font></span></a>, <a class="ref" href="glib_001.html#T38" title="GLIB_001:th.38">GLIB_001:38</a></span>;<br/><a NAME="E21:48_3_1"/>
1 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_3_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:48_3_1_2"/><span class="lab"><font color="Green" title="E70">A36</font></span>: 
1 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2_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="E1:48_3_1_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 </span><span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:48_3_1_2"><span class="lab"><font color="Green" title="E70">A36</font></span></a>, <a class="ref" href="nat_1.html#T26" title="NAT_1:th.26">NAT_1:26</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="48_3_1_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:48_3_1_2_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:48_3_1_2_1_1"/>
contradiction
 ; <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="48_3_1_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:48_3_1_2_1_2"/><span class="lab"><font color="Green" title="E71">A37</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:48_3_1_2_1_2"/><span class="lab"><font color="Green" title="E72">A38</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">d</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_3_1"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><a NAME="E3:48_3_1_2_1_2"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">e</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:48_3_1"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="txt" href="#E1:48_3_1_2_1_2"><span class="lab"><font color="Green" title="E71">A37</font></span></a>, <a class="ref" href="glib_001.html#T18" title="GLIB_001:th.18">GLIB_001:18</a></span>;<br/><span class="kw">hence </span><a NAME="E4:48_3_1_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:48_3_1"><span class="lab"><font color="Green" title="E59">A22</font></span></a>, <a class="txt" href="#E2:48_3_1_2_1_2"><span class="lab"><font color="Green" title="E72">A38</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</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>

</div><span class="kw">end;</span></div><a NAME="E22:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E70">A39</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">2 <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="chord.html#T4" title="CHORD:th.4">CHORD:4</a></span>;<br/><a NAME="E23:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A40</font></span>: 
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E24:48_3_1"/>
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:48_3_1"><span class="lab"><font color="Green" title="E65">A31</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E25:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E72">A41</font></span>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:48_3_1"><span class="lab"><font color="Green" title="E62">A25</font></span></a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</a></span>;<br/><a NAME="E26:48_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T10" target="_self" title="HELLY:th.10">Th10</a></span>;<br/><a NAME="E27:48_3_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E15:48_3_1"><span class="lab"><font color="Green" title="E66">A32</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E28:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A42</font></span>: 
<font color="Maroon" title="c6">P1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E17:48"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E23:48"><span class="lab"><font color="Green" title="E54">A17</font></span></a>, <a class="txt" href="#E26:48"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E29:48_3_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="helly.html#T10" target="_self" title="HELLY:th.10">Th10</a></span>;<br/><a NAME="E30:48_3_1"/><span class="kw">then </span>
2 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:48_3_1"><span class="lab"><font color="Green" title="E71">A40</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E31:48_3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E74">A43</font></span>: 
<font color="Maroon" title="c7">P2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">di</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="txt" href="#E17:48"><span class="lab"><font color="Green" title="E50">A13</font></span></a>, <a class="txt" href="#E16:48"><span class="lab"><font color="Green" title="E49">A12</font></span></a>, <a class="txt" href="#E26:48"><span class="lab"><font color="Green" title="E56">A19</font></span></a>, <a class="ref" href="glib_001.html#T34" title="GLIB_001:th.34">GLIB_001:34</a></span>;<br/><a NAME="E32:48_3_1"/><span class="lab"><font color="Green" title="E75">A44</font></span>: 
1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E22:48_3_1"><span class="lab"><font color="Green" title="E70">A39</font></span></a>, <a class="ref" href="finseq_3.html#T27" title="FINSEQ_3:th.27">FINSEQ_3:27</a></span>;<br/><a NAME="E33:48_3_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c15">ebi</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K8" title="GLIB_001:func.8">.cut</a> 1,<font color="Maroon" title="c17">eci</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:48_3_1"><span class="lab"><font color="Green" title="E64">A27</font></span></a>, <a class="txt" href="#E20:48_3_1"><span class="lab"><font color="Green" title="E69">A35</font></span></a>, <a class="ref" href="helly.html#T27" target="_self" title="HELLY:th.27">Th27</a></span>;<br/><span class="kw">hence </span><a NAME="E34:48_3_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:48"><span class="lab"><font color="Green" title="E40">A3</font></span></a>, <a class="txt" href="#E4:48"><span class="lab"><font color="Green" title="E41">A4</font></span></a>, <a class="txt" href="#E6:48"><span class="lab"><font color="Green" title="E43">A6</font></span></a>, <a class="txt" href="#E7:48"><span class="lab"><font color="Green" title="E44">A7</font></span></a>, <a class="txt" href="#E17:48_3_1"><span class="lab"><font color="Green" title="E67">A33</font></span></a>, <a class="txt" href="#E28:48_3_1"><span class="lab"><font color="Green" title="E73">A42</font></span></a>, <a class="txt" href="#E25:48_3_1"><span class="lab"><font color="Green" title="E72">A41</font></span></a>, <a class="txt" href="#E31:48_3_1"><span class="lab"><font color="Green" title="E74">A43</font></span></a>, <a class="txt" href="#E32:48_3_1"><span class="lab"><font color="Green" title="E75">A44</font></span></a>, <a class="ref" href="helly.html#T24" target="_self" title="HELLY:th.24">Th24</a>, <a class="ref" href="glib_001.html#T47" title="GLIB_001:th.47">GLIB_001:47</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="E2:48_3"/>
( <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  &amp; <font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:48"><span class="lab"><font color="Green" title="E46">A9</font></span></a>, <a class="txt" href="#E19:48"><span class="lab"><font color="Green" title="E52">A15</font></span></a>, <a class="ref" href="glib_001.html#T89" title="GLIB_001:th.89">GLIB_001:89</a></span>;<br/>
<a NAME="E3:48_3"/><span class="kw">then </span>
<font color="Maroon" title="c5">d</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:48_3"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">d</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c3">b</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon" title="c1">T</font> <a href="helly.html#K2" title="HELLY:func.2">.pathBetween</a> <font color="Maroon" title="c5">d</font>,<font color="Maroon" title="c4">c</font></span>)</span> <a href="glib_001.html#K13" title="GLIB_001:func.13">.vertices()</a> </span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</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>
