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<b>consider </b><font color="Maroon" title="c8">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> <b> such that </b><br/><a NAME="E2:59"/><i><font color="Green" title="E42">A1</font></i>: 
(  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c6">O</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font></span>)</span> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Olive" title="b1">t</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">t</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) &amp; ( for <font color="Olive" title="b1">nt</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> <br/> for <font color="Olive" title="b2">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> <br/> for <font color="Olive" title="b3">rts</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  st <font color="Olive" title="b3">rts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Olive" title="b2">ts</font> &amp; <font color="Olive" title="b1">nt</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b3">rts</font> holds <br/>for <font color="Olive" title="b4">x</font> being   <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of 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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>   st <font color="Olive" title="b4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Olive" title="b2">ts</font> holds <br/><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b2">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Olive" title="b1">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <font color="Olive" title="b4">x</font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="dtconstr.html#D16" target="_self" title="DTCONSTR:def.16">Def16</a></i>;<br/>
<b>reconsider </b><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/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of 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> ] <b>means </b> ex <font color="Olive" title="b1">t</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/>( $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t</font> &amp;  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Olive" title="b1">t</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="Olive" title="b1">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> );<br/>
<div><i><font color="Green" title="E43">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">s</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  implies <font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">s</font>] )</span><br/><b>assume </b><a NAME="E1:59_1"/><i><font color="Green" title="E44">A3</font></i>: 
<font color="Maroon" title="c10">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">s</font>]</span><br/><b>thus </b><a NAME="E2:59_1"/>
<font color="Maroon">S<sub>2</sub></font>[ <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">s</font>]
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><b>proof </b></a><div class="add">

<b>take </b><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c9">rtO</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font> &amp;  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Maroon" title="c11">t</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">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> )</span><br/>

<b>thus </b><a NAME="E1:59_1_1"/>
 <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c10">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">t</font>
 <b>by </b><i><a class="txt" href="#E1:59_1"><i><font color="Green" title="E44">A3</font></i></a>, <a class="txt" href="dtconstr.html#E23"><i><font color="Green" title="E17">Lm9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Maroon" title="c11">t</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">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span><br/>

<b>thus </b> <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Maroon" title="c11">t</font> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E42">A1</font></i></a></i>
<br/>.= 
<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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T47" title="FINSEQ_1:th.47">FINSEQ_1:47</a></i>
<br/>.= 
<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#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T72" title="FINSEQ_2:th.72">FINSEQ_2:72</a></i>
<br/>.= 
<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">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><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></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><b>end;</b></div>
<div><i><font color="Green" title="E44">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c10">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">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>   st <font color="Maroon" title="c10">nt</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="Olive" title="b1">ts</font> &amp; ( for <font color="Olive" title="b2">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of 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>   st <font color="Olive" title="b2">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b2">t</font>] ) holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Olive" title="b1">ts</font>]</span><br/><b>let </b><font color="Maroon" title="c11">ts</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c10">nt</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="c11">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of 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>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>] ) implies <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font>] )</span><br/><b>assume </b><a NAME="E1:59_2"/><i><font color="Green" title="E45">A5</font></i>: 
( <font color="Maroon" title="c10">nt</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="c11">ts</font> &amp; ( for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of 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>   st <font color="Olive" title="b1">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts</font> holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>] ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font>]</span><br/><b>then reconsider </b><font color="Maroon" title="c12">t'</font> = <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</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>  <b>by </b><i><a class="ref" href="dtconstr.html#D4" target="_self" title="DTCONSTR:def.4">Def4</a></i>;<br/><a NAME="E3:59_2"/><i><font color="Green" title="E46">A6</font></i>: 
( <font color="Maroon" title="c10">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> &amp; (  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c6">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> or  <a href="dtconstr.html#K1" title="DTCONSTR:func.1">roots</a> <font color="Maroon" title="c11">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default">1</span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> ) )
 <b>by </b><i><a class="txt" href="#E1:59_2"><i><font color="Green" title="E45">A5</font></i></a>, <a class="ref" href="dtconstr.html#D5" target="_self" title="DTCONSTR:def.5">Def5</a></i>;<br/><a NAME="E4:59_2"/><b>then </b>
 <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="c11">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a></i>;<br/><b>then consider </b><font color="Maroon" title="c13">x</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> <b> such that </b><br/><a NAME="E6:59_2"/><i><font color="Green" title="E47">A7</font></i>: 
( <font color="Maroon" title="c11">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c13">x</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> &amp; <font color="Maroon" title="c13">x</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>  )
 <b>by </b><i><a class="ref" href="dtconstr.html#T5" target="_self" title="DTCONSTR:th.5">Th5</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c14">x'</font> = <font color="Maroon" title="c13">x</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>  <b>by </b><i><a class="txt" href="#E6:59_2"><i><font color="Green" title="E47">A7</font></i></a></i>;<br/><a NAME="E8:59_2"/>
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts</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"><font color="Maroon" title="c13">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="#E6:59_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="finseq_1.html#T56" title="FINSEQ_1:th.56">FINSEQ_1:56</a></i>;<br/><a NAME="E9:59_2"/><b>then </b>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c11">ts</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E10:59_2"/><b>then </b><i><font color="Green" title="E48">A8</font></i>: 
 ex <font color="Olive" title="b1">t'</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="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">t'</font> &amp;  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Olive" title="b1">t'</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="Olive" title="b1">t'</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> )
 <b>by </b><i><a class="txt" href="#E1:59_2"><i><font color="Green" title="E45">A5</font></i></a></i>;<br/><a NAME="E11:59_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ;<br/><a NAME="E12:59_2"/><b>then </b><i><font color="Green" title="E49">A9</font></i>: 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">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="c14">x'</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="#E10:59_2"><i><font color="Green" title="E48">A8</font></i></a>, <a class="ref" href="dtconstr.html#D16" target="_self" title="DTCONSTR:def.16">Def16</a></i>;<br/><a NAME="E13:59_2"/>
<font color="Maroon" title="c8">f</font> <a href="finseqop.html#K5" title="FINSEQOP:func.5">*</a> <font color="Maroon" title="c11">ts</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">x</font></span>)</span></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="#E6:59_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="finseq_2.html#T39" title="FINSEQ_2:th.39">FINSEQ_2:39</a></i>;<br/><b>then </b><i><font color="Green" title="E50">A10</font></i>: <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font></span>)</span> = 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="lang1.html#K1" title="LANG1:func.1">^</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K15" title="DTCONSTR:func.15">FlattenSeq</a> <span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">x'</font></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="txt" href="#E1:59_2"><i><font color="Green" title="E45">A5</font></i></a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</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="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span>
<b>by </b><i><a class="txt" href="#E12:59_2"><i><font color="Green" title="E49">A9</font></i></a>, <a class="ref" href="dtconstr.html#T13" target="_self" title="DTCONSTR:th.13">Th13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">nt</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</font></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_1.html#T45" title="FINSEQ_1:th.45">FINSEQ_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<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="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</font></span>)</span></span>)</span> <a href="finseqop.html#K4" title="FINSEQOP:func.4">|-&gt;</a> 1</span>)</span></span>)</span> <a href="finseq_1.html#K8" title="FINSEQ_1:func.8">^</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="txt" href="#E3:59_2"><i><font color="Green" title="E46">A6</font></i></a>, <a class="ref" href="finseq_2.html#T73" title="FINSEQ_2:th.73">FINSEQ_2:73</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_1.html#K6" title="TREES_1:func.6">height</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</font></span>)</span></span>)</span> <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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="finseq_2.html#T143" title="FINSEQ_2:th.143">FINSEQ_2:143</a></i>
<br/>.= 
<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="trees_3.html#K14" title="TREES_3:func.14">^</a> <span class="p4">(<span class="default"><a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <font color="Maroon" title="c14">x'</font></span>)</span></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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
<b>by </b><i><a class="ref" href="trees_3.html#T83" title="TREES_3:th.83">TREES_3:83</a></i>
;<br/><a NAME="E15:59_2"/><i><font color="Green" title="E51">A11</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K9" title="TREES_4:func.9">-tree</a> <font color="Maroon" title="c14">x'</font></span>)</span> <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="c14">x'</font></span>)</span> &amp; <font color="Maroon" title="c12">t'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K9" title="TREES_4:func.9">-tree</a> <font color="Maroon" title="c14">x'</font> )
 <b>by </b><i><a class="txt" href="#E6:59_2"><i><font color="Green" title="E47">A7</font></i></a>, <a class="ref" href="trees_4.html#T13" title="TREES_4:th.13">TREES_4:13</a>, <a class="ref" href="trees_4.html#D5" title="TREES_4:def.5">TREES_4:def 5</a></i>;<br/><a NAME="E16:59_2"/>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">t'</font> is    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of 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>  <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ;<br/><a NAME="E17:59_2"/><b>then </b>
 <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E2:59"><i><font color="Green" title="E42">A1</font></i></a>, <a class="ref" href="dtconstr.html#D16" target="_self" title="DTCONSTR:def.16">Def16</a></i>;<br/><b>hence </b><a NAME="E18:59_2"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">nt</font> <a href="trees_4.html#K12" title="TREES_4:func.12">-tree</a> <font color="Maroon" title="c11">ts</font>]
 <b>by </b><i><a class="txt" href="#E14:59_2"><i><font color="Green" title="E50">A10</font></i></a>, <a class="txt" href="#E15:59_2"><i><font color="Green" title="E51">A11</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>
<a NAME="E6:59"/><i><font color="Green" title="E45">A12</font></i>: 
for <font color="Olive" title="b1">t</font> being  <a href="trees_2.html#NM6" title="TREES_2:NM.6">DecoratedTree</a> of 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>   st <font color="Olive" title="b1">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>  holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive" title="b1">t</font>]
 
<b>from </b><i><a class="ref" href="dtconstr.html#S7" target="_self" title="DTCONSTR:sch.7">DTCONSTR:sch 7</a>(<a class="txt" href="#E4:59"><i><font color="Green" title="E43">A2</font></i></a>, <a class="txt" href="#E5:59"><i><font color="Green" title="E44">A4</font></i></a>);<br/></i>
<b>let </b><font color="Maroon" title="c10">t</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Maroon" title="c10">t</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="c10">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span><br/>

<a NAME="E7:59"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon" title="c10">t</font>]
 
<b>by </b><i><a class="txt" href="#E6:59"><i><font color="Green" title="E45">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E8:59"/>
 <a href="dtconstr.html#K17" title="DTCONSTR:func.17">PreTraversal</a> <font color="Maroon" title="c10">t</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="c10">t</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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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