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<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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<a NAME="E1:58"/><i><font color="Green" title="E41">A1</font></i>: 
( <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> or <font color="Maroon" title="c8">nt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">O</font> )
 
<b>by </b><i><a class="txt" href="dtconstr.html#E15"><i><font color="Green" title="E10">Lm2</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>
<div><b>hereby </b> <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>,<a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><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><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">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font></span>
<div class="add"><b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</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">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font> implies <font color="Maroon" title="c9">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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><br/><b>assume </b><a NAME="E1:58_1"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/><a NAME="E2:58_1"/><b>then </b>
 ex <font color="Olive" title="b1">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>  st <br/>( <font color="Maroon" title="c9">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">TerminalString</a> <font color="Olive" title="b1">tsg</font> &amp; <font color="Olive" title="b1">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>  &amp; <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="c8">nt</font> )
 ;<br/><a NAME="E3:58_1"/><b>then </b>
<font color="Maroon" title="c9">x</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>
 <b>by </b><i><a class="ref" href="dtconstr.html#T14" target="_self" title="DTCONSTR:th.14">Th14</a></i>;<br/><b>hence </b><a NAME="E4:58_1"/>
<font color="Maroon" title="c9">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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><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"> verum</span><br/></div>
<b>end;</b></div>

<b>let </b><font color="Maroon" title="c9">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c9">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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font> )</span><br/>

<b>assume </b><a NAME="E3:58"/>
<font color="Maroon" title="c9">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="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><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font></span><br/>

<a NAME="E4:58"/><b>then </b><i><font color="Green" title="E42">A2</font></i>: 
<font color="Maroon" title="c9">x</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>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c10">prtO</font> =  <a href="dtconstr.html#K8" title="DTCONSTR:func.8">root-tree</a> <font color="Maroon" title="c6">O</font> as    <a href="subset_1.html#M1" title="SUBSET_1: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>reconsider </b><font color="Maroon" title="c11">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>reconsider </b><font color="Maroon" title="c12">srtO</font> = <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">prtO</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  <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="E8:58"/><i><font color="Green" title="E43">A3</font></i>: 
<font color="Maroon" title="c11">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>
 
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>;<br/>
<a NAME="E9:58"/><b>then </b>
<font color="Maroon" title="c7">S</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">rtO</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="dtconstr.html#E18"><i><font color="Green" title="E12">Lm4</font></i></a>, <a class="ref" href="dtconstr.html#T4" target="_self" title="DTCONSTR:th.4">Th4</a></i>;<br/>
<a NAME="E10:58"/><b>then </b><i><font color="Green" title="E44">A4</font></i>: 
<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="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">prtO</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> <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#D4" target="_self" title="DTCONSTR:def.4">Def4</a></i>;<br/>
<a NAME="E11:58"/><b>then </b>
(  <a href="dtconstr.html#K16" title="DTCONSTR:func.16">TerminalString</a> <span class="p1">(<span class="default"><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">srtO</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c7">S</font> <a href="trees_4.html#K4" title="TREES_4:func.4">-tree</a> <span class="p2"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c11">rtO</font></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span></span>)</span> <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">TerminalString</a> <font color="Maroon" title="c11">rtO</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font> )
 
<b>by </b><i><a class="txt" href="#E4:58"><i><font color="Green" title="E42">A2</font></i></a>, <a class="ref" href="dtconstr.html#T14" target="_self" title="DTCONSTR:th.14">Th14</a>, <a class="ref" href="trees_4.html#D4" title="TREES_4:def.4">TREES_4:def 4</a></i>;<br/>
<b>hence </b><a NAME="E12:58"/>
<font color="Maroon" title="c9">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K19" title="DTCONSTR:func.19">TerminalLanguage</a> <font color="Maroon" title="c8">nt</font>
 <b>by </b><i><a class="txt" href="#E1:58"><i><font color="Green" title="E41">A1</font></i></a>, <a class="txt" href="#E8:58"><i><font color="Green" title="E43">A3</font></i></a>, <a class="txt" href="#E10:58"><i><font color="Green" title="E44">A4</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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