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

<span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">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/>

<span class="kw">set </span><font color="Maroon" title="c2">E</font> =  <a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font>;<br/>
<a NAME="E1:26"/><span class="lab"><font color="Green" title="E15">A1</font></span>: 
 <a href="lang1.html#K1" title="LANG1:func.1">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#T10" target="_self" title="LANG1:th.10">Th10</a></span>;<br/>
<span class="kw">thus </span><a NAME="E2:26"/>
 <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">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>
  <span class="comment"><font color="Red">:: according to </font></span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">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/>

<span class="kw">assume </span><a NAME="E1:26_1"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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/>

<a NAME="E2:26_1"/><span class="kw">then </span>
 ex <font color="Olive" title="b1">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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> st <br/>( <font color="Maroon" title="c3">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b1">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="lang1.html#K1" title="LANG1:func.1">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> &amp; <font color="Olive" title="b1">p</font> <a href="lang1.html#R3" title="LANG1:pred.3">is_derivable_from</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY: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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )
 
;<br/>
<a NAME="E3:26_1"/><span class="kw">then </span>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E15">A1</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:26_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">let </span><font color="Maroon" title="c3">b</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E3:26"/>
<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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><br/>

<a NAME="E4:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E16">A2</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26"/>
the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D7" target="_self" title="LANG1:def.7">Def7</a></span>;<br/>
<a NAME="E6:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E17">A3</font></span>: 
<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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E7:26"/>
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#K6" title="LANG1:func.6">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D7" target="_self" title="LANG1:def.7">Def7</a></span>;<br/>
<a NAME="E8:26"/><span class="kw">then </span>
<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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E9:26"/><span class="kw">then </span>
the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a>  <a href="pre_poly.html#K2" title="PRE_POLY: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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E17">A3</font></span></a>, <a class="ref" href="lang1.html#D1" target="_self" title="LANG1:def.1">Def1</a></span>;<br/>
<a NAME="E10:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A4</font></span>: 
 <a href="pre_poly.html#K2" title="PRE_POLY: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#K6" title="LANG1:func.6">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="pre_poly.html#K3" title="PRE_POLY: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#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><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></span>;<br/>
<a NAME="E11:26"/>
 <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> 
 
;<br/>
<span class="kw">hence </span><a NAME="E12:26"/>
<font color="Maroon" title="c3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K6" title="LANG1:func.6">EmptyGrammar</a> <font color="Maroon" title="c1">a</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E15">A1</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E16">A2</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E18">A4</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
