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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</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="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font> implies  <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">T</font> =  <a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>);<br/>
<span class="kw">set </span><font color="Maroon" title="c4">I</font> = <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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <span class="kw">means </span> for <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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> $1 &amp; <font color="Olive" title="b1">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>;<br/>
<a NAME="E1:30"/><span class="lab"><font color="Green" title="E17">A1</font></span>: 
 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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <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"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D9" target="_self" title="LANG1:def.9">Def9</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:30"/>
<font color="Maroon" title="c1">a</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">b</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span><br/>

<a NAME="E3:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E18">A2</font></span>: 
 <a href="lang1.html#K1" title="LANG1:func.1">Terminals</a> <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="c2">b</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#T14" target="_self" title="LANG1:th.14">Th14</a></span>;<br/>
<span class="kw">thus </span><a NAME="E4:30"/>
 <a href="lang1.html#K3" title="LANG1:func.3">Lang</a> <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
  <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  <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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">a</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> or <font color="Maroon" title="c5">a</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_1"/>
<font color="Maroon" title="c5">a</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">a</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span><span class="kw"> such that </span><br/><a NAME="E3:30_1"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 <span class="kw">and </span><br/><a NAME="E4:30_1"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E5:30_1"/>
<font color="Maroon" title="c6">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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 ;<br/>
<a NAME="E6:30_1"/>
<font color="Maroon" title="c6">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E4:30_1"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:30_1"/>
<font color="Maroon" title="c5">a</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_1"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<a NAME="E5:30"/><span class="lab"><font color="Green" title="E19">A5</font></span>: 
 the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">a</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D9" target="_self" title="LANG1:def.9">Def9</a></span>;<br/>
<a NAME="E6:30"/><span class="lab"><font color="Green" title="E20">A6</font></span>: 
 the <a href="lang1.html#U1" title="LANG1:sel.1">Rules</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <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#K4" title="TARSKI:func.4">[</a><span class="default"><font color="Maroon" title="c1">a</font>,<span class="p3"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span>,<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#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#D9" target="_self" title="LANG1:def.9">Def9</a></span>;<br/>
<a NAME="E7:30"/><span class="kw">then </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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<a NAME="E8:30"/><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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="ref" href="lang1.html#D1" target="_self" title="LANG1:def.1">Def1</a></span>;<br/>
<a NAME="E9:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E21">A7</font></span>: 
<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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#T3" target="_self" title="LANG1:th.3">Th3</a></span>;<br/>
<a NAME="E10:30"/><span class="lab"><font color="Green" title="E22">A8</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_2"/><span class="lab"><font color="Green" title="E23">A9</font></span>: 
 for <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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>  st  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> &amp; <font color="Olive" title="b1">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  holds <br/><font color="Olive" title="b1">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c6">p</font> be   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 &amp; <font color="Maroon" title="c6">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  implies <font color="Maroon" title="c6">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:30_2"/><span class="lab"><font color="Green" title="E24">A10</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 <span class="kw">and </span><br/><a NAME="E3:30_2"/><span class="lab"><font color="Green" title="E25">A11</font></span>: 
<font color="Maroon" title="c6">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c6">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span></span><br/>

<span class="kw">consider </span><font color="Maroon" title="c7">q</font> being   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>, <font color="Maroon" title="c8">c</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E5:30_2"/><span class="lab"><font color="Green" title="E26">A12</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span>
 <span class="kw">and </span><br/><a NAME="E6:30_2"/><span class="lab"><font color="Green" title="E27">A13</font></span>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">k</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:30_2"><span class="lab"><font color="Green" title="E24">A10</font></span></a>, <a class="ref" href="trees_2.html#T3" title="TREES_2:th.3">TREES_2:3</a></span>;<br/>
<a NAME="E7:30_2"/><span class="lab"><font color="Green" title="E28">A14</font></span>: 
 <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></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="c8">c</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="finseq_1.html#T38" title="FINSEQ_1:th.38">FINSEQ_1:38</a></span>;<br/>
<a NAME="E8:30_2"/><span class="lab"><font color="Green" title="E29">A15</font></span>: 
<font color="Maroon" title="c6">p</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30_2"><span class="lab"><font color="Green" title="E25">A11</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E9:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A16</font></span>: 
<font color="Maroon" title="c7">q</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/>
<a NAME="E10:30_2"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/>
<a NAME="E11:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A17</font></span>: 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<a NAME="E12:30_2"/>
<span class="p1"><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">&lt;*</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="finseq_1.html#K9" title="FINSEQ_1:func.9">*&gt;</a></span> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E8:30_2"><span class="lab"><font color="Green" title="E29">A15</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/>
<a NAME="E13:30_2"/><span class="kw">then </span>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c8">c</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:30_2"><span class="lab"><font color="Green" title="E28">A14</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c9">c</font> = <font color="Maroon" title="c8">c</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</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="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c10">x</font> = <font color="Maroon" title="c9">c</font> as   <a href="lang1.html#NM1" title="LANG1:NM.1">Symbol</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:30_2"><span class="lab"><font color="Green" title="E31">A17</font></span></a>, <a class="ref" href="zfmisc_1.html#T31" title="ZFMISC_1:th.31">ZFMISC_1:31</a></span>;<br/>
<a NAME="E16:30_2"/><span class="lab"><font color="Green" title="E32">A18</font></span>: 
<font color="Maroon" title="c7">q</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="ref" href="finseq_1.html#T36" title="FINSEQ_1:th.36">FINSEQ_1:36</a></span>;<br/>
<a NAME="E17:30_2"/><span class="lab"><font color="Green" title="E33">A19</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>,<span class="p2"><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">&lt;*</a><span class="default"><font color="Maroon" title="c2">b</font>,<font color="Maroon" title="c1">a</font></span><a href="finseq_1.html#K10" title="FINSEQ_1:func.10">*&gt;</a></span></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:30"><span class="lab"><font color="Green" title="E20">A6</font></span></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">q</font> = <font color="Maroon" title="c7">q</font> as   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E16:30_2"><span class="lab"><font color="Green" title="E32">A18</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E19:30_2"/>
<font color="Maroon" title="c9">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 
<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="E20:30_2"/><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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <a href="lang1.html#R1" title="LANG1:pred.1">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font>, the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:30"><span class="lab"><font color="Green" title="E19">A5</font></span></a>, <a class="txt" href="#E17:30_2"><span class="lab"><font color="Green" title="E33">A19</font></span></a>, <a class="ref" href="lang1.html#D1" target="_self" title="LANG1:def.1">Def1</a></span>;<br/>
<a NAME="E21:30_2"/><span class="kw">then </span>
<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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font>, the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&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></span>;<br/>
<a NAME="E22:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A20</font></span>: 
<font color="Maroon" title="c11">q</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Maroon" title="c11">q</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</a> <span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font>, the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lang1.html#T5" target="_self" title="LANG1:th.5">Th5</a></span>;<br/>
<a NAME="E23:30_2"/>
<span class="p1"><a href="pre_poly.html#K4" title="PRE_POLY:func.4">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font>, the <a href="lang1.html#U2" title="LANG1:sel.2">InitialSym</a> of <span class="p2">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K4" title="PRE_POLY:func.4">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="pre_poly.html#K3" title="PRE_POLY:func.3">&lt;*</a><span class="default"><font color="Maroon" title="c10">x</font></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="finseq_1.html#D9" title="FINSEQ_1:def.9">FINSEQ_1:def 9</a></span>;<br/>
<a NAME="E24:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Maroon" title="c6">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="txt" href="#E5:30_2"><span class="lab"><font color="Green" title="E26">A12</font></span></a>, <a class="txt" href="#E22:30_2"><span class="lab"><font color="Green" title="E34">A20</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/>
<a NAME="E25:30_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E35">A21</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R3" title="LANG1:pred.3">is_derivable_from</a> <font color="Maroon" title="c11">q</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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#T7" target="_self" title="LANG1:th.7">Th7</a></span>;<br/>
<a NAME="E26:30_2"/>
<font color="Maroon" title="c11">q</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30_2"><span class="lab"><font color="Green" title="E30">A16</font></span></a>, <a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E27:30_2"/><span class="kw">then </span>
<font color="Maroon" title="c11">q</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="txt" href="#E1:30_2"><span class="lab"><font color="Green" title="E23">A9</font></span></a>, <a class="txt" href="#E6:30_2"><span class="lab"><font color="Green" title="E27">A13</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E28:30_2"/>
<font color="Maroon" title="c6">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="txt" href="#E25:30_2"><span class="lab"><font color="Green" title="E35">A21</font></span></a>, <a class="ref" href="lang1.html#T8" target="_self" title="LANG1:th.8">Th8</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  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="c5">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: 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="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c5">c</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  or <font color="Maroon" title="c5">c</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> )</span><br/>

<span class="kw">assume </span><a NAME="E11:30"/><span class="lab"><font color="Green" title="E23">A22</font></span>: 
<font color="Maroon" title="c5">c</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c5">c</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><br/>

<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c6">c</font> = <font color="Maroon" title="c5">c</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</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="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E13:30"/>
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="zfmisc_1.html#T7" title="ZFMISC_1:th.7">ZFMISC_1:7</a></span>;<br/>
<a NAME="E14:30"/><span class="kw">then </span>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">c</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E15:30"/><span class="kw">then </span>
<font color="Maroon" title="c6">c</font> is    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:30"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="ref" href="finseq_1.html#D4" title="FINSEQ_1:def.4">FINSEQ_1:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">c</font> as   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#D11" title="FINSEQ_1:def.11">FINSEQ_1:def 11</a></span>;<br/>
<a NAME="E17:30"/>
<font color="Maroon" title="c7">n</font> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<a NAME="E18:30"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="lang1.html#R2" title="LANG1:pred.2">==&gt;</a> <font color="Maroon" title="c7">n</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:30"><span class="lab"><font color="Green" title="E21">A7</font></span></a>, <a class="ref" href="lang1.html#T5" target="_self" title="LANG1:th.5">Th5</a></span>;<br/>
<a NAME="E19:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E24">A23</font></span>: 
<font color="Maroon" title="c7">n</font> <a href="lang1.html#R3" title="LANG1:pred.3">is_derivable_from</a> <font color="Maroon" title="c7">n</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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#T7" target="_self" title="LANG1:th.7">Th7</a></span>;<br/>
<a NAME="E20:30"/><span class="lab"><font color="Green" title="E25">A24</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="30_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">p</font> be   <a href="lang1.html#NM2" title="LANG1:NM.2">String</a> of <span class="p1">(<span class="default"><a href="lang1.html#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c8">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  implies <font color="Maroon" title="c8">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/>

<span class="kw">assume </span><a NAME="E1:30_3"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c8">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  or <font color="Maroon" title="c8">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> )</span><br/>

<a NAME="E2:30_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
;<br/>
<a NAME="E3:30_3"/><span class="kw">then </span>
<font color="Maroon" title="c8">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="finseq_1.html#T34" title="FINSEQ_1:th.34">FINSEQ_1:34</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:30_3"/>
(  not <font color="Maroon" title="c8">p</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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  or <font color="Maroon" title="c8">p</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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#T6" target="_self" title="LANG1:th.6">Th6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E21:30"/><span class="lab"><font color="Green" title="E26">A25</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="#E20:30"><span class="lab"><font color="Green" title="E25">A24</font></span></a>, <a class="txt" href="#E10:30"><span class="lab"><font color="Green" title="E22">A8</font></span></a>);<br/></span>
<a NAME="E22:30"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c7">n</font>
 
;<br/>
<a NAME="E23:30"/><span class="kw">then </span>
<font color="Maroon" title="c7">n</font> <a href="pre_poly.html#K1" title="PRE_POLY:func.1">^</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span><a href="pre_poly.html#K3" title="PRE_POLY:func.3">*&gt;</a></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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="txt" href="#E21:30"><span class="lab"><font color="Green" title="E26">A25</font></span></a>, <a class="txt" href="#E11:30"><span class="lab"><font color="Green" title="E23">A22</font></span></a></span>;<br/>
<a NAME="E24:30"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A26</font></span>: 
<font color="Maroon" title="c7">n</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</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="txt" href="#E19:30"><span class="lab"><font color="Green" title="E24">A23</font></span></a>, <a class="ref" href="lang1.html#T8" target="_self" title="LANG1:th.8">Th8</a></span>;<br/>
<a NAME="E25:30"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c6">c</font> <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"><font color="Maroon" title="c2">b</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
;<br/>
<span class="kw">hence </span><a NAME="E26:30"/>
<font color="Maroon" title="c5">c</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#K8" title="LANG1:func.8">IterGrammar</a> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:30"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E24:30"><span class="lab"><font color="Green" title="E27">A26</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
