<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">T</font>, <font color="Maroon" title="c2">T1</font> be   <a href="trees_1.html#NM1" title="TREES_1: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">P</font> being   <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</a> of <font color="Maroon" title="c1">T</font>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  holds <br/> <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Olive" title="b1">P</font>,<font color="Maroon" title="c2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b2">t1</font> where <font color="Olive" title="b2">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b3">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> holds <br/> not <font color="Olive" title="b3">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b2">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b2">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b3">s</font></span>)</span> where <font color="Olive" title="b2">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b3">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">P</font> </span>}</span> </span><br/><b>let </b><font color="Maroon" title="c3">P</font> be    <a href="trees_1.html#M4" title="TREES_1:mode.4">AntiChain_of_Prefixes</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"> ( <font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  implies  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/>





<b>assume </b><a NAME="E1:8"/><i><font color="Green" title="E9">A1</font></i>: 
<font color="Maroon" title="c3">P</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> </span><br/>

<a NAME="E2:8"/><b>then </b><i><font color="Green" title="E10">A2</font></i>: 
 <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="trees_a.html#T2" target="_self" title="TREES_A:th.2">Th2</a></i>;<br/>
<b>thus </b><a NAME="E3:8"/>
 <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
  <i><font color="Red">:: according to </font></i><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/>

<b>assume </b><a NAME="E1:8_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> </span><br/>

<a NAME="E2:8_1"/><b>then </b><i><font color="Green" title="E11">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="trees_a.html#E1:8"><i><font color="Green" title="E9">A1</font></i></a>, <a class="ref" href="trees_a.html#T2" target="_self" title="TREES_A:th.2">Th2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> or  not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_1"/><i><font color="Green" title="E12">A4</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/><div class="add"><a NAME="E2:8_1_1_1_1"/>
<font color="Maroon" title="c3">P</font> <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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="trees_a.html#T6" target="_self" title="TREES_A:th.6">Th6</a></i>;<br/><b>hence </b><a NAME="E3:8_1_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 <b>by </b><i><a class="txt" href="trees_a.html#E1:8_1_1_1_1"><i><font color="Green" title="E12">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2"/><i><font color="Green" title="E12">A5</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:8_1_1_1_2_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="trees_a.html#E2:8_1"><i><font color="Green" title="E11">A3</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_1_1_1"/><i><font color="Green" title="E13">A6</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/><div class="add"><a NAME="E2:8_1_1_1_2_1_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1_1_1_2_1_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:8_1_1_1_2_1_1_1_1"/><i><font color="Green" title="E14">A7</font></i>: 
not <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<a NAME="E2:8_1_1_1_2_1_1_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="trees_1.html#NR3" title="TREES_1:NR.3">is_a_proper_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="subset_1.html#K6" title="SUBSET_1:func.6">\</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P</font>
 
<b>by </b><i><a class="ref" href="trees_a.html#T5" target="_self" title="TREES_A:th.5">Th5</a></i>;<br/>
<b>hence </b><a NAME="E3:8_1_1_1_2_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="trees_a.html#E1:8_1_1_1_2"><i><font color="Green" title="E12">A5</font></i></a>, <a class="txt" href="trees_a.html#E1:8_1_1_1_2_1_1_1"><i><font color="Green" title="E13">A6</font></i></a>, <a class="txt" href="trees_a.html#E1:8_1_1_1_2_1_1_1_1"><i><font color="Green" title="E14">A7</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:8_1_1_1_2_1_1_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="8_1_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:8_1_1_1_2_1_1_2"/>
<font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )</span><br/><div class="add"><b>hence </b><a NAME="E2:8_1_1_1_2_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:8_1_1_1_2"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  or <font color="Maroon" title="c4">x</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="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:8_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E4:8"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">t1</font> where <font color="Olive" title="b1">t1</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font> : for <font color="Olive" title="b2">p</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> holds <br/> not <font color="Olive" title="b2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">is_a_prefix_of</a> <font color="Olive" title="b1">t1</font> </span>}</span>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b1">p</font> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <font color="Olive" title="b2">s</font></span>)</span> where <font color="Olive" title="b1">p</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c1">T</font>, <font color="Olive" title="b2">s</font> is    <a href="trees_1.html#M1" title="TREES_1:mode.1">Element</a> of <font color="Maroon" title="c2">T1</font> : <font color="Olive" title="b1">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="trees_a.html#K1" title="TREES_A:func.1">tree</a> <font color="Maroon" title="c1">T</font>,<font color="Maroon" title="c3">P</font>,<font color="Maroon" title="c2">T1</font>
 <b>by </b><i><a class="txt" href="trees_a.html#E2:8"><i><font color="Green" title="E10">A2</font></i></a>, <a class="ref" href="trees_a.html#T3" target="_self" title="TREES_A:th.3">Th3</a>, <a class="ref" href="xboole_1.html#T9" title="XBOOLE_1:th.9">XBOOLE_1:9</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
