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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E1:30"/><i><font color="Green">E15</font></i>: 
 <a href="lang1.html#K5" target="_self">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="ref" href="lang1.html#T10" target="_self">Th10</a></i>;<br/>
<b>thus </b><a NAME="E2:30"/>
 <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="lang1.html#NM2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green">E16</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="lang1.html#K5" target="_self">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> )
 ;<br/>
<a NAME="E4:30_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lang1.html#E1:30"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lang1.html#E3:30_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="xboole_1.html#T3">XBOOLE_1:3</a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 
<b>by </b><i><a class="txt" href="lang1.html#E3:30_1"><i><font color="Green">E16</font></i></a>, <a class="ref" href="finseq_1.html#T27">FINSEQ_1:27</a></i>;<br/>
<b>hence </b><a NAME="E6:30_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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

<a NAME="E3:30"/><i><font color="Green">E16</font></i>: 
( the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; the <a href="lang1.html#U1" target="_self">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span> )
 
<b>by </b><i><a class="ref" href="lang1.html#D7" target="_self">Def7</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E4:30"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K1">}</a></span>
 ;<br/>

<a NAME="E5:30"/><b>then </b><i><font color="Green">E17</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<a href="xboole_0.html#K1">{}</a> </span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> the <a href="lang1.html#U1" target="_self">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K1">{}</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> &amp;  <a href="relat_1.html#K2">rng</a> <a href="xboole_0.html#K1">{}</a>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="lang1.html#E3:30"><i><font color="Green">E16</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:30"/><b>then </b>
the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lang1.html#R1" target="_self">==&gt;</a>  <a href="lang1.html#K2" target="_self">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lang1.html#D1" target="_self">Def1</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
<span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span> <a href="lang1.html#R2" target="_self">==&gt;</a>  <a href="lang1.html#K2" target="_self">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="lang1.html#T3" target="_self">Th3</a></i>;<br/>
<a NAME="E8:30"/><b>then </b>
 <a href="lang1.html#K2" target="_self">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="lang1.html#R3" target="_self">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" target="_self">&lt;*</a><span class="default">the <a href="lang1.html#U2" target="_self">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="lang1.html#K3" target="_self">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="lang1.html#T7" target="_self">Th7</a></i>;<br/>
<b>hence </b><a NAME="E9:30"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="lang1.html#K7" target="_self">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" target="_self">EmptyGrammar</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="lang1.html#E1:30"><i><font color="Green">E15</font></i></a>, <a class="txt" href="lang1.html#E5:30"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
