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

<span class="kw">let </span><font color="Maroon" title="c1">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a> holds  <span class="p1">(<span class="default"><font color="Olive" title="b1">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font> is <font color="Olive" title="b1">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">S</font> be   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</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">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</a> </span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">T</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">SS</font> =  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">L</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">S1</font> = 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">f</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">ff</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">FF</font> = <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a> ;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E27">B10</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">LS</font> =  <a href="fomodel1.html#K16" title="FOMODEL1:func.16">LettersOf</a> <font color="Maroon" title="c2">S</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c12">L1</font> = 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">LS</font>;<br/>
<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <span class="kw">means </span><span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1 is <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</a> ;<br/>
<a NAME="E3:69"/><span class="lab"><font color="Green" title="E28">B1</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="69_1"><span class="kw">proof </span></a><div class="add">

<span class="p1">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">LS</font></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> = 
1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">LS</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T3" title="FOMODEL0:th.3">FOMODEL0:3</a></span>
<br/>.= 
1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">LS</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">L11</font> = 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <font color="Maroon" title="c11">LS</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ;<br/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> = 
 <a href="fomodel1.html#K26" title="FOMODEL1:func.26">AtomicTermsOf</a> <font color="Maroon" title="c2">S</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D30" target="_self" title="FOMODEL1:def.30">DefTerm</a></span>
<br/>.= 
<font color="Maroon" title="c13">L11</font>

;<br/>
<span class="kw">hence </span><a NAME="E4:69_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <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"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E4:69"/><span class="lab"><font color="Green" title="E29">B2</font></span>: 
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c13">m</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">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="c13">m</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:69_2"/><span class="lab"><font color="Green" title="E30">Z5</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">m</font>]
 ; <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="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]</span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c14">mm</font> = <font color="Maroon" title="c13">m</font>, <font color="Maroon" title="c15">mm1</font> = <font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D12" title="ORDINAL1:def.12">ORDINAL1:def 12</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c16">t1</font>, <font color="Maroon" title="c17">t2</font>, <font color="Maroon" title="c18">w1</font>, <font color="Maroon" title="c19">w2</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="c16">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c17">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c19">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">t1</font>,<font color="Maroon" title="c18">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t2</font>,<font color="Maroon" title="c19">w2</font>) implies ( <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w2</font> ) )</span><br/><span class="kw">assume </span><a NAME="E1:69_2_1"/><span class="lab"><font color="Green" title="E31">Z0</font></span>: 
( <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c17">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span></span>)</span> <a href="xboole_0.html#K3" title="XBOOLE_0:func.3">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c19">w2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</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"> ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">t1</font>,<font color="Maroon" title="c18">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t2</font>,<font color="Maroon" title="c19">w2</font>) implies ( <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w2</font> ) )</span><br/><a NAME="E2:69_2_1"/>
( <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">mm1</font> &amp;  not <font color="Maroon" title="c16">t1</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> &amp; <font color="Maroon" title="c17">t2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c15">mm1</font> &amp;  not <font color="Maroon" title="c17">t2</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="txt" href="#E1:69_2_1"><span class="lab"><font color="Green" title="E31">Z0</font></span></a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c20">t11</font> = <font color="Maroon" title="c16">t1</font>, <font color="Maroon" title="c21">t22</font> = <font color="Maroon" title="c17">t2</font> as  <font color="Maroon" title="c14">mm</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1 <a href="fomodel1.html#V14" title="FOMODEL1:attr.14">-termal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#D33" target="_self" title="FOMODEL1:def.33">DefTermal</a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c22">t111</font> = <font color="Maroon" title="c20">t11</font>, <font color="Maroon" title="c23">t222</font> = <font color="Maroon" title="c21">t22</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <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/><span class="kw">consider </span><font color="Maroon" title="c24">T1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> <span class="kw"> such that </span><br/><a NAME="E6:69_2_1"/><span class="lab"><font color="Green" title="E32">C1</font></span>: 
( <font color="Maroon" title="c24">T1</font> is  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Maroon" title="c20">t11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T3" target="_self" title="FOMODEL1:th.3">Lm333</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c25">T2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">mm</font></span>)</span> <a href="fomodel0.html#K2" title="FOMODEL0:func.2">*</a> <span class="kw"> such that </span><br/><a NAME="E8:69_2_1"/><span class="lab"><font color="Green" title="E33">C2</font></span>: 
( <font color="Maroon" title="c25">T2</font> is  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span>)</span> <a href="card_1.html#V3" title="CARD_1:attr.3">-element</a>  &amp; <font color="Maroon" title="c21">t22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel1.html#T3" target="_self" title="FOMODEL1:th.3">Lm333</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c26">T11</font> = <font color="Maroon" title="c24">T1</font>, <font color="Maroon" title="c27">T22</font> = <font color="Maroon" title="c25">T2</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">mm</font> <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/><span class="kw">reconsider </span><font color="Maroon" title="c28">w11</font> = <font color="Maroon" title="c18">w1</font>, <font color="Maroon" title="c29">w22</font> = <font color="Maroon" title="c19">w2</font> as    <a href="finseq_2.html#M2" title="FINSEQ_2:mode.2">Element</a> of <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</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="#E1:69_2_1"><span class="lab"><font color="Green" title="E31">Z0</font></span></a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c30">head1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font>, <font color="Maroon" title="c31">head2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font>, <font color="Maroon" title="c32">tail1</font> = <font color="Maroon" title="c28">w11</font>, <font color="Maroon" title="c33">tail2</font> = <font color="Maroon" title="c29">w22</font> as    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <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/><span class="kw">assume </span><a NAME="E12:69_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c16">t1</font>,<font color="Maroon" title="c18">w1</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c17">t2</font>,<font color="Maroon" title="c19">w2</font>)
 ; <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="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w2</font> )</span><br/><a NAME="E13:69_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c22">t111</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c28">w11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c23">t222</font>,<font color="Maroon" title="c29">w22</font>)
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a></span>;<br/><a NAME="E14:69_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c28">w11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">t222</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c29">w22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_2_1"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a></span>;<br/><a NAME="E15:69_2_1"/><span class="kw">then </span>
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c28">w11</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font></span>)</span></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <font color="Maroon" title="c29">w22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:69_2_1"><span class="lab"><font color="Green" title="E33">C2</font></span></a>, <a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><a NAME="E16:69_2_1"/><span class="kw">then </span>
<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font></span>)</span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">head2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c33">tail2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T32" title="FINSEQ_1:th.32">FINSEQ_1:32</a></span>;<br/><a NAME="E17:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">Z1</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c31">head2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c33">tail2</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a></span>;<br/><a NAME="E18:69_2_1"/>
1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span>
 ;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c34">S1</font> = 1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default">1 <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> ;<br/><a NAME="E20:69_2_1"/><span class="lab"><font color="Green" title="E35">C4</font></span>: 
<font color="Maroon" title="c34">S1</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a>  <a href="fomodel0.html#V2" title="FOMODEL0:attr.2">-unambiguous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#D3" title="FOMODEL0:def.3">FOMODEL0:def 3</a></span>;<br/><a NAME="E21:69_2_1"/>
<font color="Maroon" title="c34">S1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c34">S1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_2.html#T134" title="FINSEQ_2:th.134">FINSEQ_2:134</a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>;<br/><a NAME="E22:69_2_1"/><span class="kw">then </span>
( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">S1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c34">S1</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c31">head2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c33">tail2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font></span>)</span>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K20" title="FOMODEL1:func.20">-cons</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span>,<span class="p1">(<span class="default"><font color="Maroon" title="c31">head2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c33">tail2</font></span>)</span>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:69_2_1"><span class="lab"><font color="Green" title="E34">Z1</font></span></a>, <a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a></span>;<br/><a NAME="E23:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E36">Z3</font></span>: 
( <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <span class="p1"><a href="matrix_2.html#K4" title="MATRIX_2:func.4">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font></span>)</span></span><a href="matrix_2.html#K4" title="MATRIX_2:func.4">*&gt;</a></span> &amp; <font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <font color="Maroon" title="c31">head2</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c33">tail2</font> &amp; <font color="Maroon" title="c30">head1</font> <a href="monoid_0.html#K1" title="MONOID_0:func.1">^</a> <font color="Maroon" title="c32">tail1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="binop_1.html#K1" title="BINOP_1:func.1">.</a> (<font color="Maroon" title="c30">head1</font>,<font color="Maroon" title="c32">tail1</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E20:69_2_1"><span class="lab"><font color="Green" title="E35">C4</font></span></a>, <a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a>, <a class="ref" href="fomodel0.html#D9" title="FOMODEL0:def.9">FOMODEL0:def 9</a></span>;<br/><a NAME="E24:69_2_1"/><span class="lab"><font color="Green" title="E37">C6</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">t22</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:69_2_1"><span class="lab"><font color="Green" title="E36">Z3</font></span></a>, <a class="ref" href="finseq_1.html#T76" title="FINSEQ_1:th.76">FINSEQ_1:76</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c35">n</font> =  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span>;<br/><a NAME="E25:69_2_1"/>
(  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c26">T11</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c27">T22</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_2_1"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="txt" href="#E24:69_2_1"><span class="lab"><font color="Green" title="E37">C6</font></span></a>, <a class="txt" href="#E8:69_2_1"><span class="lab"><font color="Green" title="E33">C2</font></span></a>, <a class="ref" href="card_1.html#D7" title="CARD_1:def.7">CARD_1:def 7</a></span>;<br/><a NAME="E26:69_2_1"/><span class="kw">then </span>
( <font color="Maroon" title="c26">T11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> &amp; <font color="Maroon" title="c27">T22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span> &amp; <font color="Maroon" title="c26">T11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> &amp; <font color="Maroon" title="c27">T22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E27">B10</font></span></a>, <a class="ref" href="finseq_2.html#T133" title="FINSEQ_2:th.133">FINSEQ_2:133</a></span>;<br/><a NAME="E27:69_2_1"/><span class="kw">then </span>
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c26">T11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c27">T22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" title="FUNCT_1:def.6">FUNCT_1:def 6</a></span>;<br/><a NAME="E28:69_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E38">C9</font></span>: 
( <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span></span>)</span> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a></span>)</span> &amp; <font color="Maroon" title="c28">w11</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <font color="Maroon" title="c29">w22</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="finseq_2.html#K3" title="FINSEQ_2:func.3">*</a>  &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c24">T1</font></span>)</span>,<font color="Maroon" title="c28">w11</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a></span>)</span> <a href="binop_1.html#K4" title="BINOP_1:func.4">.</a> (<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c25">T2</font></span>)</span>,<font color="Maroon" title="c29">w22</font>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E23:69_2_1"><span class="lab"><font color="Green" title="E36">Z3</font></span></a>, <a class="ref" href="fomodel0.html#T4" title="FOMODEL0:th.4">FOMODEL0:4</a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></span>;<br/><a NAME="E29:69_2_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> is  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <a href="fomodel0.html#V3" title="FOMODEL0:attr.3">-prefix</a>  &amp; ( <span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font> is  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <a href="fomodel0.html#V3" title="FOMODEL0:attr.3">-prefix</a>  implies <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> is  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <a href="fomodel0.html#V3" title="FOMODEL0:attr.3">-prefix</a>  ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E30">Z5</font></span></a>, <a class="ref" href="fomodel1.html#D28" target="_self" title="FOMODEL1:def.28">DefSPrefix</a></span>;<br/><a NAME="E30:69_2_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="fomodel0.html#K8" title="FOMODEL0:func.8">-multiCat</a></span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="int_2.html#K1" title="INT_2:func.1">abs</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K19" title="FOMODEL1:func.19">ar</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c20">t11</font></span>)</span></span>)</span></span>)</span> <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">m</font></span>)</span></span>)</span> is <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a>  <a href="fomodel0.html#V2" title="FOMODEL0:attr.2">-unambiguous</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#D3" title="FOMODEL0:def.3">FOMODEL0:def 3</a></span>;<br/><span class="kw">hence </span><a NAME="E31:69_2_1"/>
( <font color="Maroon" title="c16">t1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">t2</font> &amp; <font color="Maroon" title="c18">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">w2</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:69_2_1"><span class="lab"><font color="Green" title="E32">C1</font></span></a>, <a class="txt" href="#E8:69_2_1"><span class="lab"><font color="Green" title="E33">C2</font></span></a>, <a class="txt" href="#E23:69_2_1"><span class="lab"><font color="Green" title="E36">Z3</font></span></a>, <a class="txt" href="#E28:69_2_1"><span class="lab"><font color="Green" title="E38">C9</font></span></a>, <a class="ref" href="fomodel0.html#D9" title="FOMODEL0:def.9">FOMODEL0:def 9</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="E4:69_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is <span class="p1">(<span class="default"><a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span> <a href="monoid_0.html#K11" title="MONOID_0:func.11">-concatenation</a>  <a href="fomodel0.html#V2" title="FOMODEL0:attr.2">-unambiguous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#D9" title="FOMODEL0:def.9">FOMODEL0:def 9</a></span>;<br/>
<a NAME="E5:69_2"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1</span>)</span> is  <a href="fomodel1.html#K15" title="FOMODEL1:func.15">AllSymbolsOf</a> <font color="Maroon" title="c2">S</font> <a href="fomodel0.html#V3" title="FOMODEL0:attr.3">-prefix</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#D3" title="FOMODEL0:def.3">FOMODEL0:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c13">m</font> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> 1]
 ; <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="E5:69"/>
 for <font color="Olive" title="b1">m</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">m</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="nat_1.html#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E28">B1</font></span></a>, <a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E29">B2</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E6:69"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K28" title="FOMODEL1:func.28">-termsOfMaxDepth</a></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">m</font> is <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#V12" title="FOMODEL1:attr.12">-prefix</a> 
 ; <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>
