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

<b>let </b><font color="Maroon" title="c1">a</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">  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<b>set </b><font color="Maroon" title="c2">E</font> =  <a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font>;<br/>
<a NAME="E1:30"/><i><font color="Green" title="E15">A1</font></i>: 
 <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="ref" href="lang1.html#T10" target="_self" title="LANG1:th.10">Th10</a></i>;<br/>
<b>thus </b><a NAME="E2:30"/>
 <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><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" title="c3">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="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="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )</span><br/>

<b>assume </b><a NAME="E1:30_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span><br/>

<b>then consider </b><font color="Maroon" title="c4">p</font> being   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span><b> such that </b><br/><a NAME="E3:30_1"/><i><font color="Green" title="E16">A2</font></i>: 
( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lang1.html#K5" title="LANG1:func.5">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c4">p</font> <a href="lang1.html#R3" title="LANG1:pred.3">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default">the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span> )
 ;<br/>
<a NAME="E4:30_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E3:30_1"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<a NAME="E5:30_1"/><b>then </b>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="#E3:30_1"><i><font color="Green" title="E16">A2</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:30_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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>

<a NAME="E3:30"/><i><font color="Green" title="E16">A3</font></i>: 
( the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> &amp; the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">a</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="ref" href="lang1.html#D7" target="_self" title="LANG1:def.7">Def7</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">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="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="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> or <font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E4:30"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </span><a href="tarski.html#K1" title="TARSKI:func.1">}</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="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/>

<a NAME="E5:30"/><b>then </b><i><font color="Green" title="E17">A4</font></i>: 
( <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">a</font>,<a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> </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 <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp;  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 
<b>by </b><i><a class="txt" href="#E3:30"><i><font color="Green" title="E16">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<a NAME="E6:30"/><b>then </b>
the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<b>by </b><i><a class="ref" href="lang1.html#D1" target="_self" title="LANG1:def.1">Def1</a></i>;<br/>
<a NAME="E7:30"/><b>then </b>
 <a href="lang1.html#K2" title="LANG1:func.2">&lt;*&gt;</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="lang1.html#R3" title="LANG1:pred.3">is_derivable_from</a> <span class="p1"><a href="lang1.html#K3" title="LANG1:func.3">&lt;*</a><span class="default">the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="lang1.html#K3" title="LANG1:func.3">*&gt;</a></span>
 
<b>by </b><i><a class="ref" href="lang1.html#T3" target="_self" title="LANG1:th.3">Th3</a>, <a class="ref" href="lang1.html#T7" target="_self" title="LANG1:th.7">Th7</a></i>;<br/>
<b>hence </b><a NAME="E8:30"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K7" title="LANG1:func.7">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K10" title="LANG1:func.10">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E1:30"><i><font color="Green" title="E15">A1</font></i></a>, <a class="txt" href="#E5:30"><i><font color="Green" title="E17">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/>


</div>
