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

<b>let </b><font color="Maroon" title="c1">S1</font>, <font color="Maroon" title="c2">S2</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">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> ;<br/>

<b>assume </b><a NAME="E1:3_1_2"/><i><font color="Green" title="E5">A25</font></i>: 
(  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c1">S1</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="Maroon" title="c1">S1</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="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">S1</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="Olive" title="b1">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>assume </b><a NAME="E2:3_1_2"/><i><font color="Green" title="E6">A26</font></i>: 
(  <a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c2">S2</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="Maroon" title="c2">S2</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="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="c2">S2</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="Olive" title="b1">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/>

<i><font color="Green" title="E7">A27</font></i>: the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">S1</font> = 
<span class="p1">(<span class="default"><a href="dtconstr.html#K6" title="DTCONSTR:func.6">Terminals</a> <font color="Maroon" title="c1">S1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="dtconstr.html#K7" title="DTCONSTR:func.7">NonTerminals</a> <font color="Maroon" title="c1">S1</font></span>)</span>
<b>by </b><i><a class="ref" href="lang1.html#T1" title="LANG1:th.1">LANG1:1</a></i>
<br/>.= 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">S2</font>
<b>by </b><i><a class="txt" href="scm_comp.html#E1:3_1_2"><i><font color="Green" title="E5">A25</font></i></a>, <a class="txt" href="scm_comp.html#E2:3_1_2"><i><font color="Green" title="E6">A26</font></i></a>, <a class="ref" href="lang1.html#T1" title="LANG1:th.1">LANG1:1</a></i>
;<br/>
<a NAME="E4:3_1_2"/>
the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">S2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="3_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</font>, <font color="Maroon" title="c4">b</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="relat_1.html#D2" title="RELAT_1:def.2">RELAT_1:def 2</a><br/>

<b>set </b><font color="Maroon" title="c5">p1</font> = the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">S1</font>;<br/>
<b>set </b><font color="Maroon" title="c6">p2</font> = the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">S2</font>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:3_1_2_2_1"/><i><font color="Green" title="E8">A28</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">S1</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c3">a</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c4">b</font> as    <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 <font color="Maroon" title="c1">S1</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="scm_comp.html#E1:3_1_2_2_1"><i><font color="Green" title="E8">A28</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/><a NAME="E4:3_1_2_2_1"/><i><font color="Green" title="E9">A29</font></i>: 
<font color="Maroon" title="c7">l</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c8">r</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E1:3_1_2_2_1"><i><font color="Green" title="E8">A28</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/><b>then consider </b><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">S1</font><b> such that </b><br/><a NAME="E6:3_1_2_2_1"/><i><font color="Green" title="E10">A30</font></i>: 
<font color="Maroon" title="c8">r</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="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="bintree1.html#D4" title="BINTREE1:def.4">BINTREE1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c7">l</font>, <font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c13">x2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_2"><i><font color="Green" title="E7">A27</font></i></a></i>;<br/><a NAME="E8:3_1_2_2_1"/>
<font color="Maroon" title="c11">l</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#E1:3_1_2"><i><font color="Green" title="E5">A25</font></i></a>, <a class="txt" href="scm_comp.html#E4:3_1_2_2_1"><i><font color="Green" title="E9">A29</font></i></a>, <a class="txt" href="scm_comp.html#E6:3_1_2_2_1"><i><font color="Green" title="E10">A30</font></i></a></i>;<br/><a NAME="E9:3_1_2_2_1"/><b>then </b>
<font color="Maroon" title="c11">l</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="c12">x1</font>,<font color="Maroon" title="c13">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#E2:3_1_2"><i><font color="Green" title="E6">A26</font></i></a></i>;<br/><b>hence </b><a NAME="E10:3_1_2_2_1"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">S2</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E6:3_1_2_2_1"><i><font color="Green" title="E10">A30</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/></div>
<b>end;</b></div>

<b>assume </b><a NAME="E2:3_1_2_2"/><i><font color="Green" title="E8">A31</font></i>: 
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c2">S2</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c7">l</font> = <font color="Maroon" title="c3">a</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">S2</font> <b>by </b><i><a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">r</font> = <font color="Maroon" title="c4">b</font> as    <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 <font color="Maroon" title="c2">S2</font> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_2_2"><i><font color="Green" title="E8">A31</font></i></a>, <a class="ref" href="zfmisc_1.html#T106" title="ZFMISC_1:th.106">ZFMISC_1:106</a></i>;<br/>
<a NAME="E5:3_1_2_2"/><i><font color="Green" title="E9">A32</font></i>: 
<font color="Maroon" title="c7">l</font> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <font color="Maroon" title="c8">r</font>
 
<b>by </b><i><a class="txt" href="scm_comp.html#E2:3_1_2_2"><i><font color="Green" title="E8">A31</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c10">x2</font> being   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c2">S2</font><b> such that </b><br/><a NAME="E7:3_1_2_2"/><i><font color="Green" title="E10">A33</font></i>: 
<font color="Maroon" title="c8">r</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="c9">x1</font>,<font color="Maroon" title="c10">x2</font></span><a href="lang1.html#K4" title="LANG1:func.4">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="bintree1.html#D4" title="BINTREE1:def.4">BINTREE1:def 4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">l</font> = <font color="Maroon" title="c7">l</font>, <font color="Maroon" title="c12">x1</font> = <font color="Maroon" title="c9">x1</font>, <font color="Maroon" title="c13">x2</font> = <font color="Maroon" title="c10">x2</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <font color="Maroon" title="c1">S1</font> <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_2"><i><font color="Green" title="E7">A27</font></i></a></i>;<br/>
<a NAME="E9:3_1_2_2"/>
<font color="Maroon" title="c11">l</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_2"><i><font color="Green" title="E6">A26</font></i></a>, <a class="txt" href="scm_comp.html#E5:3_1_2_2"><i><font color="Green" title="E9">A32</font></i></a>, <a class="txt" href="scm_comp.html#E7:3_1_2_2"><i><font color="Green" title="E10">A33</font></i></a></i>;<br/>
<a NAME="E10:3_1_2_2"/><b>then </b>
<font color="Maroon" title="c11">l</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="c12">x1</font>,<font color="Maroon" title="c13">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#E1:3_1_2"><i><font color="Green" title="E5">A25</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:3_1_2_2"/>
<span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <font color="Maroon" title="c1">S1</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E7:3_1_2_2"><i><font color="Green" title="E10">A33</font></i></a>, <a class="ref" href="lang1.html#D1" title="LANG1:def.1">LANG1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:3_1_2"/>
<font color="Maroon" title="c1">S1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">S2</font>
 <b>by </b><i><a class="txt" href="scm_comp.html#E3:3_1_2"><i><font color="Green" title="E7">A27</font></i></a></i>;<br/>


</div>
