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<b>thus </b><a NAME="E1:30"/>
 <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E22"><i><font color="Green" title="E16">Lm8</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D6" target="_self" title="DTCONSTR:def.6">DTCONSTR:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  is <a href="dtconstr.html#V2" title="DTCONSTR:attr.2">with_nonterminals</a> &amp;  <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  is <a href="dtconstr.html#V3" title="DTCONSTR:attr.3">with_useful_nonterminals</a> )</span><br/>

<b>thus </b><a NAME="E2:30"/>
 <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 <b>by </b><i><a class="txt" href="dtconstr.html#E25"><i><font color="Green" title="E19">Lm11</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D7" target="_self" title="DTCONSTR:def.7">DTCONSTR:def 7</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  is <a href="dtconstr.html#V3" title="DTCONSTR:attr.3">with_useful_nonterminals</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c5">rO</font> =  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c2">O</font> as    <a href="dtconstr.html#M1" title="DTCONSTR:mode.1">Element</a> of <font color="Maroon" title="c4">TSPN</font> <b>by </b><i><a class="ref" href="dtconstr.html#D4" target="_self" title="DTCONSTR:def.4">Def4</a>, <a class="txt" href="dtconstr.html#E22"><i><font color="Green" title="E16">Lm8</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">p</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">rO</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> as    <a href="trees_4.html#M1" title="TREES_4:mode.1">FinSequence</a> of <font color="Maroon" title="c4">TSPN</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c7">p</font> = <font color="Maroon" title="c6">p</font> as    <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/>
<b>let </b><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> ; <i><font color="Red">:: according to </font></i><a class="ref" href="dtconstr.html#D8" target="_self" title="DTCONSTR:def.8">DTCONSTR:def 8</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <a href="dtconstr.html#K5" title="DTCONSTR:func.5">PeanoNat</a>  implies  ex <font color="Olive" title="b1">p</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="c8">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">p</font> )</span><br/>

<b>assume </b><a NAME="E6:30"/>
<font color="Maroon" title="c8">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</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">  ex <font color="Olive" title="b1">p</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="c8">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">p</font></span><br/>

<a NAME="E7:30"/><b>then </b><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">S</font>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E27"><i><font color="Green" title="E21">Lm13</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c7">p</font>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c8">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="c7">p</font></span><br/>

<a NAME="E8:30"/><i><font color="Green" title="E28">A2</font></i>: 
(  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c5">rO</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 &amp;  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> 1 )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T55" title="FINSEQ_1:th.55">FINSEQ_1:55</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c9">i</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font> implies  ex <font color="Olive" title="b1">rO</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">rO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rO</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>  ) )</span><br/><b>assume </b><a NAME="E1:30_1"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c9">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">rO</font> being  <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> st <br/>( <font color="Olive" title="b1">rO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">rO</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>  )</span><br/><b>reconsider </b><font color="Maroon" title="c10">rO</font> = <font color="Maroon" title="c5">rO</font> as   <a href="trees_2.html#NM4" title="TREES_2:NM.4">DecoratedTree</a> ;<br/><b>take </b><font color="Maroon" title="c11">rO</font> = <font color="Maroon" title="c10">rO</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c11">rO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">rO</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>  )</span><br/><a NAME="E3:30_1"/>
( <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">O</font> )
 <b>by </b><i><a class="txt" href="#E8:30"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="#E1:30_1"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="finseq_1.html#T4" title="FINSEQ_1:th.4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E4:30_1"/>
( <font color="Maroon" title="c11">rO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> &amp; <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c2">O</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">rO</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>  )
 <b>by </b><i><a class="ref" href="finseq_1.html#T57" title="FINSEQ_1:th.57">FINSEQ_1:57</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>
<b>hence </b><a NAME="E10:30"/>
<font color="Maroon" title="c8">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="c7">p</font>
 <b>by </b><i><a class="txt" href="#E7:30"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="#E8:30"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="dtconstr.html#E18"><i><font color="Green" title="E12">Lm4</font></i></a>, <a class="ref" href="trees_3.html#D18" title="TREES_3:def.18">TREES_3:def 18</a></i>; <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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