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

<span class="kw">let </span><font color="Maroon" title="c8">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) &amp; ( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  ) )</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">rtO</font> =  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  ;<br/>
<a NAME="E2:58"/>
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_1.html#T79" title="TREES_1:th.79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/>
<a NAME="E3:58"/><span class="kw">then </span>
 <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c9">rtO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="dtconstr.html#T16" target="_self" title="DTCONSTR:th.16">Th16</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E41">A1</font></span>:  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c9">rtO</font> = 
<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>

<br/>.= 
<span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></span>
;<br/>
<span class="kw">thus </span><a NAME="E5:58"/>
( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="58_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_2"/><span class="lab"><font color="Green" title="E42">A2</font></span>: 
<font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<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="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> implies <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><span class="kw">assume </span><a NAME="E1:58_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</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="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">tsg</font> being    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <span class="kw"> such that </span><br/><a NAME="E3:58_2_1"/><span class="lab"><font color="Green" title="E43">A3</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c11">tsg</font>
 <span class="kw">and </span><br/><a NAME="E4:58_2_1"/><span class="lab"><font color="Green" title="E44">A4</font></span>: 
<font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 <span class="kw">and </span><br/><a NAME="E5:58_2_1"/><span class="lab"><font color="Green" title="E45">A5</font></span>: 
<font color="Maroon" title="c11">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a></span>;<br/><a NAME="E6:58_2_1"/>
<font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_2_1"><span class="lab"><font color="Green" title="E44">A4</font></span></a>, <a class="txt" href="#E5:58_2_1"><span class="lab"><font color="Green" title="E45">A5</font></span></a>, <a class="ref" href="dtconstr.html#T9" target="_self" title="DTCONSTR:th.9">Th9</a></span>;<br/><span class="kw">hence </span><a NAME="E7:58_2_1"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E3:58_2_1"><span class="lab"><font color="Green" title="E43">A3</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>

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> )</span><br/>

<span class="kw">assume </span><a NAME="E3:58_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font></span><br/>

<a NAME="E4:58_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A6</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:58_2"/>
<font color="Maroon" title="c9">rtO</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:58_2"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58"><span class="lab"><font color="Green" title="E41">A1</font></span></a>, <a class="txt" href="#E1:58_2"><span class="lab"><font color="Green" title="E42">A2</font></span></a>, <a class="txt" href="#E4:58_2"><span class="lab"><font color="Green" title="E43">A6</font></span></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>

<span class="kw">assume </span><a NAME="E6:58"/><span class="lab"><font color="Green" title="E42">A7</font></span>: 
<font color="Maroon" title="c8">nt</font> <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">  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/>

<div><span class="kw">hereby </span> <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<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">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font></span>
<div class="add"><span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> implies <font color="Maroon" title="c10">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"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  )</span><br/><span class="kw">assume </span><a NAME="E1:58_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</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="c10">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"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> </span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">tsg</font> being    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <span class="kw"> such that </span><br/><a NAME="E3:58_3"/><span class="lab"><font color="Green" title="E43">A8</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c11">tsg</font>
 <span class="kw">and </span><br/><a NAME="E4:58_3"/><span class="lab"><font color="Green" title="E44">A9</font></span>: 
<font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 <span class="kw">and </span><br/><a NAME="E5:58_3"/><span class="lab"><font color="Green" title="E45">A10</font></span>: 
<font color="Maroon" title="c11">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E42">A7</font></span></a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c12">ts</font> being    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <span class="kw"> such that </span><br/><a NAME="E7:58_3"/><span class="lab"><font color="Green" title="E46">A11</font></span>: 
<font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c12">ts</font>
 <span class="kw">and </span><br/><a NAME="E8:58_3"/><span class="lab"><font color="Green" title="E47">A12</font></span>: 
<font color="Maroon" title="c7">S</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c12">ts</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_3"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="txt" href="#E5:58_3"><span class="lab"><font color="Green" title="E45">A10</font></span></a>, <a class="ref" href="dtconstr.html#T10" target="_self" title="DTCONSTR:th.10">Th10</a></span>;<br/><a NAME="E9:58_3"/>
(  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c12">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> or  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c12">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default">1</span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:58_3"><span class="lab"><font color="Green" title="E47">A12</font></span></a>, <a class="ref" href="dtconstr.html#D5" target="_self" title="DTCONSTR:def.5">Def5</a></span>;<br/><a NAME="E10:58_3"/><span class="kw">then </span>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c12">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">t</font> being    <a href="trees_3.html#M5" title="TREES_3:mode.5">Element</a> of  <a href="trees_3.html#K5" title="TREES_3:func.5">FinTrees</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <span class="kw"> such that </span><br/><a NAME="E12:58_3"/><span class="lab"><font color="Green" title="E48">A13</font></span>: 
<font color="Maroon" title="c12">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">t</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E13:58_3"/>
<font color="Maroon" title="c13">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dtconstr.html#T5" target="_self" title="DTCONSTR:th.5">Th5</a></span>;<br/><a NAME="E14:58_3"/>
<font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> <a href="trees_4.html#K9" title="TREES_4:func.9">-tree</a> <font color="Maroon" title="c13">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:58_3"><span class="lab"><font color="Green" title="E46">A11</font></span></a>, <a class="txt" href="#E12:58_3"><span class="lab"><font color="Green" title="E48">A13</font></span></a>, <a class="ref" href="trees_4.html#D5" title="TREES_4:def.5">TREES_4:def 5</a></span>;<br/><a NAME="E15:58_3"/><span class="kw">then </span>
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="trees_3.html#K14" title="TREES_3:func.14">^</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">t</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T13" title="TREES_4:th.13">TREES_4:13</a></span>;<br/><a NAME="E16:58_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A14</font></span>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">tsg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">t</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T83" title="TREES_3:th.83">TREES_3:83</a></span>;<br/><a NAME="E17:58_3"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c11">tsg</font></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:58_3"><span class="lab"><font color="Green" title="E43">A8</font></span></a>, <a class="txt" href="#E4:58_3"><span class="lab"><font color="Green" title="E44">A9</font></span></a>, <a class="ref" href="dtconstr.html#T16" target="_self" title="DTCONSTR:th.16">Th16</a></span>;<br/><span class="kw">hence </span><a NAME="E18:58_3"/>
<font color="Maroon" title="c10">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"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:58_3"><span class="lab"><font color="Green" title="E49">A14</font></span></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>

<span class="kw">let </span><font color="Maroon" title="c10">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c10">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"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span>  or <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font> )</span><br/>

<span class="kw">assume </span><a NAME="E8:58"/>
<font color="Maroon" title="c10">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"><span class="p3">(<span class="default"><font color="Olive" title="b1">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p3"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span>)</span> where <font color="Olive" title="b1">n</font> is    <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>  : <font color="Olive" title="b1">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c11">n</font> being    <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="E10:58"/><span class="lab"><font color="Green" title="E43">A15</font></span>: 
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E11:58"/><span class="lab"><font color="Green" title="E44">A16</font></span>: 
<font color="Maroon" title="c11">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>2</sub></font>[   <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">means </span>( $1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  ex <font color="Olive" title="b1">tsg</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Olive" title="b1">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">$1 <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> ) );<br/>
<a NAME="E12:58"/><span class="lab"><font color="Green" title="E45">A17</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
;<br/>
<div><span class="lab"><font color="Green" title="E46">A18</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="58_4"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c12">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">n</font>] implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/><span class="kw">assume </span><a NAME="E1:58_4"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">n</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">S<sub>2</sub></font>[<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/><span class="kw">thus </span><a NAME="E2:58_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
  <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="58_4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:58_4_1"/>
<font color="Maroon" title="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">  ex <font color="Olive" title="b1">tsg</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Olive" title="b1">tsg</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_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="E1:58_4_1_1"/>
( <font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:58_4_1_1_1"/>
<font color="Maroon" title="c12">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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">  ex <font color="Olive" title="b1">tsg</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Olive" title="b1">tsg</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">tsg</font> being    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <span class="kw"> such that </span><br/><a NAME="E3:58_4_1_1_1"/>
<font color="Maroon" title="c13">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font>
 <span class="kw">and </span><br/><a NAME="E4:58_4_1_1_1"/><span class="lab"><font color="Green" title="E48">A20</font></span>: 
 <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c13">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:58_4"><span class="lab"><font color="Green" title="E47">A19</font></span></a></span>;<br/><a NAME="E5:58_4_1_1_1"/>
 <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c13">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p3">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">tsg</font></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="dtconstr.html#T16" target="_self" title="DTCONSTR:th.16">Th16</a></span>;<br/><a NAME="E6:58_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E49">A21</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">tsg</font></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:58_4_1_1_1"><span class="lab"><font color="Green" title="E48">A20</font></span></a>, <a class="ref" href="finseq_1.html#T46" title="FINSEQ_1:th.46">FINSEQ_1:46</a></span>;<br/><a NAME="E7:58_4_1_1_1"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><a NAME="E8:58_4_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E50">A22</font></span>: 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">tsg</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">n</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58_4_1_1_1"><span class="lab"><font color="Green" title="E49">A21</font></span></a>, <a class="ref" href="finseq_1.html#D18" title="FINSEQ_1:def.18">FINSEQ_1:def 18</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c14">tsg9</font> = <font color="Maroon" title="c7">S</font> <a href="dtconstr.html#K10" title="DTCONSTR:func.10">-tree</a> <font color="Maroon" title="c13">tsg</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="c14">tsg9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c14">tsg9</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/><a NAME="E9:58_4_1_1_1"/><span class="lab"><font color="Green" title="E51">A23</font></span>: 
<font color="Maroon" title="c14">tsg9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">tsg</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#D5" title="TREES_4:def.5">TREES_4:def 5</a></span>;<br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">tsg9</font></span>)</span> = 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K14" title="TREES_3:func.14">^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">tsg</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T13" title="TREES_4:th.13">TREES_4:13</a></span>
<br/>.= 
<font color="Maroon" title="c12">n</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="#E8:58_4_1_1_1"><span class="lab"><font color="Green" title="E50">A22</font></span></a>, <a class="ref" href="trees_3.html#T83" title="TREES_3:th.83">TREES_3:83</a></span>
;<br/><span class="kw">hence </span><a NAME="E11:58_4_1_1_1"/>
( <font color="Maroon" title="c14">tsg9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c14">tsg9</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:58_4_1_1_1"><span class="lab"><font color="Green" title="E51">A23</font></span></a>, <a class="ref" href="dtconstr.html#T16" target="_self" title="DTCONSTR:th.16">Th16</a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="58_4_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:58_4_1_1_2"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<font color="Maroon" title="c12">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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">  ex <font color="Olive" title="b1">tsg</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Olive" title="b1">tsg</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/><div class="add"><span class="kw">take </span><font color="Maroon" title="c13">tsg9</font> = <font color="Maroon" title="c7">S</font> <a href="dtconstr.html#K10" title="DTCONSTR:func.10">-tree</a> <font color="Maroon" title="c9">rtO</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="c13">tsg9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c13">tsg9</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/><a NAME="E2:58_4_1_1_2"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
<font color="Maroon" title="c13">tsg9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c9">rtO</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#D5" title="TREES_4:def.5">TREES_4:def 5</a></span>;<br/> <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c13">tsg9</font></span>)</span> = 
 <a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p1">(<span class="default"><a href="trees_3.html#K14" title="TREES_3:func.14">^</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">rtO</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_4.html#T13" title="TREES_4:th.13">TREES_4:13</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p2">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c9">rtO</font></span>)</span></span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
<span class="kw">by</span> <span class="lab"><a class="ref" href="trees_3.html#T83" title="TREES_3:th.83">TREES_3:83</a></span>
<br/>.= 
<font color="Maroon" title="c12">n</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="#E1:58_4_1_1_2"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="trees_1.html#T79" title="TREES_1:th.79">TREES_1:79</a>, <a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:58_4_1_1_2"/>
( <font color="Maroon" title="c13">tsg9</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Maroon" title="c13">tsg9</font> <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="c12">n</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:58_4_1_1_2"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="dtconstr.html#T16" target="_self" title="DTCONSTR:th.16">Th16</a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</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></div><span class="kw">end;</span></div>
<a NAME="E14:58"/>
for <font color="Olive" title="b1">n</font> being   <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>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">n</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E12:58"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E13:58"><span class="lab"><font color="Green" title="E46">A18</font></span></a>);<br/></span>
<a NAME="E15:58"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">tsg</font> being   <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  st <br/>( <font color="Olive" title="b1">tsg</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp;  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">PreTraversal</a> <font color="Olive" title="b1">tsg</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">n</font> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:58"><span class="lab"><font color="Green" title="E44">A16</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E16:58"/>
<font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">PreTraversalLanguage</a> <font color="Maroon" title="c8">nt</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:58"><span class="lab"><font color="Green" title="E42">A7</font></span></a>, <a class="txt" href="#E10:58"><span class="lab"><font color="Green" title="E43">A15</font></span></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>
