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

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b>$1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>;<br/>
<b>consider </b><font color="Maroon" title="c1">G</font> being  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lang1.html#V1" title="LANG1:attr.1">strict</a> <a href="bintree1.html#V3" title="BINTREE1:attr.3">binary</a>  <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</a> <b> such that </b><br/><a NAME="E2:3_1_1"/><i><font color="Green" title="E5">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>and </b><br/><a NAME="E3:3_1_1"/><i><font color="Green" title="E6">A2</font></i>: 
for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> iff <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font>] )
 <b>from </b><i><a class="ref" href="bintree1.html#S1" title="BINTREE1:sch.1">BINTREE1:sch 1</a>();<br/></i>
<a NAME="E4:3_1_1"/><i><font color="Green" title="E7">A3</font></i>: 
 <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> : for <font color="Olive" title="b2">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds not <font color="Olive" title="b1">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">tnt</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lang1.html#D2" title="LANG1:def.2">LANG1:def 2</a></i>;<br/>
<a NAME="E5:3_1_1"/><i><font color="Green" title="E8">A4</font></i>: 
 <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_1_1"/>
 <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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><br/>

<b>assume </b><a NAME="E1:3_1_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">t</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:3_1_1_1_1"/><i><font color="Green" title="E9">A5</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> &amp; ( for <font color="Olive" title="b1">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>  holds not <font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b1">tnt</font> ) )
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:3_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>assume </b><a NAME="E4:3_1_1_1_1"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ;<br/>

<a NAME="E5:3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="txt" href="scm_comp.html#E3:3_1_1_1_1"><i><font color="Green" title="E9">A5</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E6:3_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:3_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1_1_1"><i><font color="Green" title="E9">A5</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c2">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><br/>

<b>assume </b><a NAME="E2:3_1_1_1"/><i><font color="Green" title="E9">A6</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">t</font> = <font color="Maroon" title="c2">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<b>assume </b><a NAME="E4:3_1_1_1"/>
not <font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">tnt</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E6:3_1_1_1"/><i><font color="Green" title="E10">A7</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c4">tnt</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E4:3_1_1"><i><font color="Green" title="E7">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E8:3_1_1_1"/><i><font color="Green" title="E11">A8</font></i>: 
<font color="Maroon" title="c4">tnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E6:3_1_1_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="ref" href="bintree1.html#D4" title="BINTREE1:def.4">BINTREE1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c7">k</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:3_1_1_1"/><i><font color="Green" title="E12">A9</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 <b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_1_1"><i><font color="Green" title="E9">A6</font></i></a>, <a class="ref" href="ami_2.html#D2" title="AMI_2:def.2">AMI_2:def 2</a></i>;<br/>
<b>set </b><font color="Maroon" title="c8">m</font> = <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1;<br/>
<a NAME="E11:3_1_1_1"/>
0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">i</font> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  : <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">*</a> <font color="Maroon" title="c7">k</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 </span>}</span> 
 
;<br/>
<a NAME="E12:3_1_1_1"/><b>then </b><i><font color="Green" title="E13">A10</font></i>: 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">x</font>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E10:3_1_1_1"><i><font color="Green" title="E12">A9</font></i></a>, <a class="ref" href="axioms.html#T30" title="AXIOMS:th.30">AXIOMS:30</a></i>;<br/>
<a NAME="E13:3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scm_comp.html#E6:3_1_1_1"><i><font color="Green" title="E10">A7</font></i></a>, <a class="txt" href="scm_comp.html#E8:3_1_1_1"><i><font color="Green" title="E11">A8</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E15:3_1_1_1"/><i><font color="Green" title="E14">A11</font></i>: 
( <font color="Maroon" title="c9">x1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 1 &amp; <font color="Maroon" title="c10">x2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 5 &amp; <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T103" title="ZFMISC_1:th.103">ZFMISC_1:103</a></i>;<br/>
<a NAME="E16:3_1_1_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><span class="p2"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>,<span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c9">x1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E15:3_1_1_1"><i><font color="Green" title="E14">A11</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:3_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="scm_comp.html#E12:3_1_1_1"><i><font color="Green" title="E13">A10</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:3_1_1"/><i><font color="Green" title="E9">A12</font></i>: 
 <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">t</font> where <font color="Olive" title="b1">t</font> is   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> :  ex <font color="Olive" title="b2">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Olive" title="b1">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b2">tnt</font> </span>}</span> 
 
<b>by </b><i><a class="ref" href="lang1.html#D3" title="LANG1:def.3">LANG1:def 3</a></i>;<br/>
<a NAME="E7:3_1_1"/><i><font color="Green" title="E10">A13</font></i>: 
 <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:3_1_1_2"/>
 <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c2">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><br/>

<b>assume </b><a NAME="E1:3_1_1_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c3">t</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E3:3_1_1_2_1"/><i><font color="Green" title="E11">A14</font></i>: 
( <font color="Maroon" title="c2">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">t</font> &amp;  ex <font color="Olive" title="b1">tnt</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> st <font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Olive" title="b1">tnt</font> )
 <b>by </b><i><a class="txt" href="scm_comp.html#E6:3_1_1"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c4">tnt</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a><b> such that </b><br/><a NAME="E5:3_1_1_2_1"/><i><font color="Green" title="E12">A15</font></i>: 
<font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c4">tnt</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1_2_1"><i><font color="Green" title="E11">A14</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c5">x1</font>, <font color="Maroon" title="c6">x2</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font><b> such that </b><br/><a NAME="E7:3_1_1_2_1"/><i><font color="Green" title="E13">A16</font></i>: 
<font color="Maroon" title="c4">tnt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c5">x1</font>,<font color="Maroon" title="c6">x2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E5:3_1_1_2_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="ref" href="bintree1.html#D4" title="BINTREE1:def.4">BINTREE1:def 4</a></i>;<br/>
<b>thus </b><a NAME="E8:3_1_1_2_1"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scm_comp.html#E3:3_1_1_2_1"><i><font color="Green" title="E11">A14</font></i></a>, <a class="txt" href="scm_comp.html#E5:3_1_1_2_1"><i><font color="Green" title="E12">A15</font></i></a>, <a class="txt" href="scm_comp.html#E7:3_1_1_2_1"><i><font color="Green" title="E13">A16</font></i></a></i>;<br/>


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

<b>let </b><font color="Maroon" title="c2">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><br/>

<b>assume </b><a NAME="E2:3_1_1_2"/><i><font color="Green" title="E11">A17</font></i>: 
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c3">t</font> = <font color="Maroon" title="c2">x</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E4:3_1_1_2"/>
<font color="Maroon" title="c3">t</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">t</font>,<font color="Maroon" title="c3">t</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scm_comp.html#E2:3_1_1_2"><i><font color="Green" title="E11">A17</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:3_1_1_2"/>
<font color="Maroon" title="c2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E6:3_1_1"><i><font color="Green" title="E9">A12</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:3_1_1"/><i><font color="Green" title="E11">A18</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="dtconstr.html#V1" title="DTCONSTR:attr.1">with_terminals</a>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E5:3_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="ref" href="dtconstr.html#D6" title="DTCONSTR:def.6">DTCONSTR:def 6</a></i>;<br/>
<a NAME="E9:3_1_1"/><i><font color="Green" title="E12">A19</font></i>: 
<font color="Maroon" title="c1">G</font> is <a href="dtconstr.html#V2" title="DTCONSTR:attr.2">with_nonterminals</a>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E7:3_1_1"><i><font color="Green" title="E10">A13</font></i></a>, <a class="ref" href="dtconstr.html#D7" title="DTCONSTR:def.7">DTCONSTR:def 7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">nt</font> be   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font>;<br/><b>assume </b><a NAME="E1:3_1_1_3"/><i><font color="Green" title="E13">A20</font></i>: 
<font color="Maroon" title="c2">nt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K6" title="LANG1:func.6">NonTerminals</a> <font color="Maroon" title="c1">G</font>
 ;<br/><a NAME="E2:3_1_1_3"/><i><font color="Green" title="E14">A21</font></i>: 
(  <a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  &amp;  <a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  )
 <b>by </b><i><a class="ref" href="ami_3.html#D2" title="AMI_3:def.2">AMI_3:def 2</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c3">d0</font> =  <a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 0, <font color="Maroon" title="c4">d1</font> =  <a href="ami_3.html#K15" title="AMI_3:func.15">dl.</a> 1 as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_1"><i><font color="Green" title="E5">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E4:3_1_1_3"/>
(  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c3">d0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c1">G</font> &amp;  <a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c1">G</font> )
 <b>by </b><i><a class="txt" href="scm_comp.html#E5:3_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="scm_comp.html#E2:3_1_1_3"><i><font color="Green" title="E14">A21</font></i></a>, <a class="ref" href="dtconstr.html#D4" title="DTCONSTR:def.4">DTCONSTR:def 4</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c5">p</font> = <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c3">d0</font></span>)</span>,<span class="p2">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c4">d1</font></span>)</span></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="dtconstr.html#K4" title="DTCONSTR:func.4">TS</a> <font color="Maroon" title="c1">G</font> <b>by </b><i><a class="ref" href="finseq_2.html#T15" title="FINSEQ_2:th.15">FINSEQ_2:15</a></i>;<br/><i><font color="Green" title="E15">A22</font></i>:  <a href="trees_3.html#K16" title="TREES_3:func.16">roots</a> <font color="Maroon" title="c5">p</font> = 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c3">d0</font></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> </span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c4">d1</font></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> </span>)</span></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="dtconstr.html#T6" title="DTCONSTR:th.6">DTCONSTR:6</a></i>
<br/>.= 
<span class="p1"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="trees_4.html#K2" title="TREES_4:func.2">root-tree</a> <font color="Maroon" title="c3">d0</font></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> </span>)</span>,<font color="Maroon" title="c4">d1</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span>
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>
<br/>.= 
<span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c3">d0</font>,<font color="Maroon" title="c4">d1</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
<b>by </b><i><a class="ref" href="trees_4.html#T3" title="TREES_4:th.3">TREES_4:3</a></i>
;<br/><b>take </b><font color="Maroon" title="c6">p</font> = <font color="Maroon" title="c5">p</font>;<br/><b>thus </b><a NAME="E7:3_1_1_3"/>
<font color="Maroon" title="c2">nt</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> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scm_comp.html#E7:3_1_1"><i><font color="Green" title="E10">A13</font></i></a>, <a class="txt" href="scm_comp.html#E1:3_1_1_3"><i><font color="Green" title="E13">A20</font></i></a>, <a class="txt" href="scm_comp.html#E6:3_1_1_3"><i><font color="Green" title="E15">A22</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E11:3_1_1"/><b>then </b>
<font color="Maroon" title="c1">G</font> is <a href="dtconstr.html#V3" title="DTCONSTR:attr.3">with_useful_nonterminals</a>
 
<b>by </b><i><a class="ref" href="dtconstr.html#D8" title="DTCONSTR:def.8">DTCONSTR:def 8</a></i>;<br/>
<b>hence </b><a NAME="E12:3_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="lang1.html#V1" title="LANG1:attr.1">strict</a> <a href="dtconstr.html#V1" title="DTCONSTR:attr.1">with_terminals</a> <a href="dtconstr.html#V2" title="DTCONSTR:attr.2">with_nonterminals</a> <a href="dtconstr.html#V3" title="DTCONSTR:attr.3">with_useful_nonterminals</a> <a href="bintree1.html#V3" title="BINTREE1:attr.3">binary</a>  <a href="lang1.html#L1" title="LANG1:struct.1">DTConstrStr</a>  st <br/>(  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="ami_2.html#K4" title="AMI_2:func.4">SCM-Data-Loc</a>  &amp;  <a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> &amp; ( for <font color="Olive" title="b2">x</font>, <font color="Olive" title="b3">y</font>, <font color="Olive" title="b4">z</font> being  <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Olive">b<sub>1</sub></font> holds <br/> ( <font color="Olive" title="b2">x</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="lang1.html#K4" title="LANG1:func.4">&lt;*</a><span class="default"><font color="Olive" title="b3">y</font>,<font color="Olive" title="b4">z</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span> iff <font color="Olive" title="b2">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default">1,5</span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span> ) ) )
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_1"><i><font color="Green" title="E6">A2</font></i></a>, <a class="txt" href="scm_comp.html#E5:3_1_1"><i><font color="Green" title="E8">A4</font></i></a>, <a class="txt" href="scm_comp.html#E7:3_1_1"><i><font color="Green" title="E10">A13</font></i></a>, <a class="txt" href="scm_comp.html#E8:3_1_1"><i><font color="Green" title="E11">A18</font></i></a>, <a class="txt" href="scm_comp.html#E9:3_1_1"><i><font color="Green" title="E12">A19</font></i></a></i>;<br/>


</div>
