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

<span class="kw">let </span><font color="Maroon" title="c1">U</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <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">  for <font color="Olive" title="b1">S</font> being   <a href="fomodel1.html#NM2" title="FOMODEL1:NM.2">Language</a><br/>  for <font color="Olive" title="b2">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Olive" title="b1">S</font><br/>  for <font color="Olive" title="b3">I1</font>, <font color="Olive" title="b4">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Olive" title="b1">S</font>  st <font color="Olive" title="b3">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b2">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Olive" title="b1">S</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b3">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b2">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b2">phi</font></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">  for <font color="Olive" title="b1">phi</font> being   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b2">I1</font>, <font color="Olive" title="b3">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b2">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b2">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">phi</font> be   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>; <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">I1</font>, <font color="Olive" title="b2">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font></span><br/>





<span class="kw">set </span><font color="Maroon" title="c4">O</font> =  <a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c5">II</font> = <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">a</font> =  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">E</font> =  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">S</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">F</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K22" title="FOMODEL1:func.22">-firstChar</a> ;<br/>
<span class="kw">set </span><font color="Maroon" title="c9">C</font> = <font color="Maroon" title="c2">S</font> <a href="fomodel1.html#K21" title="FOMODEL1:func.21">-multiCat</a> ;<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> for <font color="Olive" title="b1">I1</font>, <font color="Olive" title="b2">I2</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font><br/>  for <font color="Olive" title="b3">phi</font> being  $1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>  st <font color="Olive" title="b1">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Olive" title="b1">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b3">phi</font>;<br/>
<a NAME="E1:156"/><span class="lab"><font color="Green" title="E71">B0</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="156_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">I1</font>, <font color="Maroon" title="c11">I2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font>; <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">phi</font> being   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font></span><br/><span class="kw">let </span><font color="Maroon" title="c12">phi</font> be   <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</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="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> implies <font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c13">phi1</font> = <font color="Maroon" title="c12">phi</font> as   <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">assume </span><a NAME="E2:156_1"/>
<font color="Maroon" title="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c12">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></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="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font></span><br/>

<a NAME="E3:156_1"/><span class="kw">then </span>
( <font color="Maroon" title="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp;  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c2">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  the <a href="fomodel1.html#U1" title="FOMODEL1:sel.1">adicity</a> of <font color="Maroon" title="c2">S</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> &amp;  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">S</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="fomodel2.html#K34" title="FOMODEL2:func.34">TheEqSymbOf</a> <font color="Maroon" title="c2">S</font> )
 
;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c14">phi2</font> being   <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font><span class="kw"> such that </span><br/><a NAME="E5:156_1"/><span class="lab"><font color="Green" title="E72">C0</font></span>: 
( <font color="Maroon" title="c14">phi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">phi1</font> &amp; <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c14">phi2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K14" title="FOMODEL2:func.14">-AtomicEval</a> <font color="Maroon" title="c13">phi1</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="fomodel3.html#E84"><span class="lab"><font color="Green" title="E69">Lm52</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:156_1"/>
<font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c12">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:156_1"><span class="lab"><font color="Green" title="E72">C0</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><span class="kw">end;</span></div>
<a NAME="E2:156"/><span class="lab"><font color="Green" title="E72">B1</font></span>: 
 for <font color="Olive" title="b1">n</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">n</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c10">n</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="c10">n</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/>

<span class="kw">assume </span><a NAME="E1:156_2"/><span class="lab"><font color="Green" title="E73">C1</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c10">n</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="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/>

<span class="kw">let </span><font color="Maroon" title="c11">I1</font>, <font color="Maroon" title="c12">I2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font>; <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">phi</font> being  <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font>  st <font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Olive" title="b1">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> holds <br/><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Olive" title="b1">phi</font></span><br/><span class="kw">let </span><font color="Maroon" title="c13">phi</font> be  <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</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="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> implies <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> )</span><br/>



<span class="kw">assume </span><a NAME="E2:156_2"/><span class="lab"><font color="Green" title="E74">C0</font></span>: 
<font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></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="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font></span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:156_2_1"/>
( (  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp;  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  ) or ( <font color="Maroon" title="c13">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  &amp;  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  ) or <font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1_1"><span class="kw">suppose </span></a><a NAME="E1:156_2_1_1"/>
(  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  &amp;  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</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="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">phii</font> = <font color="Maroon" title="c13">phi</font> as   non  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</a>  <font color="Maroon" title="c10">n</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>   non  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c15">X</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span>;<br/><span class="kw">set </span><font color="Maroon" title="c16">s</font> = <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="c14">phii</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c17">h</font> =  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c14">phii</font>, <font color="Maroon" title="c18">t</font> =  <a href="fomodel2.html#K42" title="FOMODEL2:func.42">tail</a> <font color="Maroon" title="c14">phii</font> as  <font color="Maroon" title="c10">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/><font color="Maroon" title="c14">phii</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&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="c14">phii</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T23" title="FOMODEL2:th.23">FOMODEL2:23</a></span>
<br/>.= 
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&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="c14">phii</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t</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="E5:156_2_1_1"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t</font></span>)</span> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">h</font> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c18">t</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T29" title="FINSEQ_1:th.29">FINSEQ_1:29</a>, <a class="ref" href="finseq_1.html#T30" title="FINSEQ_1:th.30">FINSEQ_1:30</a></span>;<br/><a NAME="E6:156_2_1_1"/><span class="kw">then </span>
(  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font> &amp;  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">t</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font> )
 <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/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">rh</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span>, <font color="Maroon" title="c20">rt</font> = <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c18">t</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</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"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T26" title="XBOOLE_1:th.26">XBOOLE_1:26</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c21">v1</font> = <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font>;<br/><span class="kw">set </span><font color="Maroon" title="c22">v2</font> = <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font>;<br/><span class="kw">set </span><font color="Maroon" title="c23">h1</font> = <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font>;<br/><span class="kw">set </span><font color="Maroon" title="c24">h2</font> = <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font>;<br/><span class="kw">set </span><font color="Maroon" title="c25">t1</font> = <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font>;<br/><span class="kw">set </span><font color="Maroon" title="c26">t2</font> = <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font>;<br/><span class="lab"><font color="Green" title="E75">DD0</font></span>: <font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c19">rh</font> = 
<font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">rh</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c19">rh</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">rh</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156_2"><span class="lab"><font color="Green" title="E74">C0</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
;<br/><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">rt</font> = 
<font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">rt</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c20">rt</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">rt</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156_2"><span class="lab"><font color="Green" title="E74">C0</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
;<br/><a NAME="E10:156_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">D1</font></span>: 
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_2"><span class="lab"><font color="Green" title="E73">C1</font></span></a></span>;<br/><a NAME="E11:156_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="xboolean.html#K9" title="XBOOLEAN:func.9">'nor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="xboole_0.html#K5" title="XBOOLE_0:func.5">\+\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="xboolean.html#K9" title="XBOOLEAN:func.9">'nor'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font></span>)</span></span>)</span> <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="E12:156_2_1_1"/><span class="kw">then </span>
( <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="xboolean.html#K9" title="XBOOLEAN:func.9">'nor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font></span>)</span> &amp; <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c17">h</font></span>)</span> <a href="xboolean.html#K9" title="XBOOLEAN:func.9">'nor'</a> <span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c18">t</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T29" title="FOMODEL0:th.29">FOMODEL0:29</a></span>;<br/><span class="kw">hence </span><a NAME="E13:156_2_1_1"/>
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156_2"><span class="lab"><font color="Green" title="E73">C1</font></span></a>, <a class="txt" href="#E8:156_2_1_1"><span class="lab"><font color="Green" title="E75">DD0</font></span></a>, <a class="txt" href="#E10:156_2_1_1"><span class="lab"><font color="Green" title="E76">D1</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><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1_2"><span class="kw">suppose </span></a><a NAME="E1:156_2_1_2"/>
( <font color="Maroon" title="c13">phi</font> is  <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  &amp;  not <font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</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="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font></span><br/><div class="add"><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c14">phii</font> = <font color="Maroon" title="c13">phi</font> as   <a href="fomodel2.html#V4" title="FOMODEL2:attr.4">wff</a>   <a href="fomodel2.html#V5" title="FOMODEL2:attr.5">exal</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="kw">set </span><font color="Maroon" title="c15">l</font> = <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="c14">phii</font>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c16">h</font> =  <a href="fomodel2.html#K40" title="FOMODEL2:func.40">head</a> <font color="Maroon" title="c14">phii</font> as  <font color="Maroon" title="c10">n</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/><span class="lab"><font color="Green" title="E75">D0</font></span>: <font color="Maroon" title="c14">phii</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&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="c14">phii</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="finseq_1.html#K7" title="FINSEQ_1:func.7">^</a> <span class="p1">(<span class="default"><a href="fomodel2.html#K41" title="FOMODEL2:func.41">tail</a> <font color="Maroon" title="c14">phii</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel2.html#T23" title="FOMODEL2:th.23">FOMODEL2:23</a></span>
<br/>.= 
<span class="p1"><a href="fomodel2.html#K31" title="FOMODEL2:func.31">&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="c14">phii</font></span>)</span></span><a href="fomodel2.html#K31" title="FOMODEL2:func.31">*&gt;</a></span> <a href="fomodel2.html#K30" title="FOMODEL2:func.30">^</a> <font color="Maroon" title="c16">h</font>

;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">rh</font> =  <a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h</font> as   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="ref" href="finseq_1.html#T30" title="FINSEQ_1:th.30">FINSEQ_1:30</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="156_2_1_2_2"><span class="kw">now </span></a><div class="add"><div><span class="kw">hereby </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="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 implies <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )</span>
<div class="add"><span class="kw">assume </span><a NAME="E1:156_2_1_2_2_1"/>
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">U</font><span class="kw"> such that </span><br/><a NAME="E3:156_2_1_2_2_1"/><span class="lab"><font color="Green" title="E76">E0</font></span>: 
<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#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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c11">I1</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156_2_1_2"><span class="lab"><font color="Green" title="E75">D0</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">f</font> = <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">I1u</font> = (<span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c11">I1</font>, <font color="Maroon" title="c21">I2u</font> = (<span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c12">I2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font> ;<br/><font color="Maroon" title="c20">I1u</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">rh</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">rh</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156_2"><span class="lab"><font color="Green" title="E74">C0</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">rh</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c21">I2u</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></span>
;<br/><a NAME="E6:156_2_1_2_2_1"/><span class="kw">then </span>
<font color="Maroon" title="c21">I2u</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156_2_1_2_2_1"><span class="lab"><font color="Green" title="E76">E0</font></span></a>, <a class="txt" href="#E1:156_2"><span class="lab"><font color="Green" title="E73">C1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E7:156_2_1_2_2_1"/>
<font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156_2_1_2"><span class="lab"><font color="Green" title="E75">D0</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</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">assume </span><a NAME="E2:156_2_1_2_2"/>
<font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ; <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="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">u</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">U</font><span class="kw"> such that </span><br/><a NAME="E4:156_2_1_2_2"/><span class="lab"><font color="Green" title="E76">E0</font></span>: 
<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#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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c12">I2</font></span>)</span> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156_2_1_2"><span class="lab"><font color="Green" title="E75">D0</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</a></span>;<br/><span class="kw">set </span><font color="Maroon" title="c19">f</font> = <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p1">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c20">I1u</font> = (<span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c11">I1</font>, <font color="Maroon" title="c21">I2u</font> = (<span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span>,<font color="Maroon" title="c18">u</font>) <a href="fomodel2.html#K37" title="FOMODEL2:func.37">ReassignIn</a> <font color="Maroon" title="c12">I2</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</font> ;<br/><font color="Maroon" title="c20">I1u</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="fomodel0.html#K13" title="FOMODEL0:func.13">null</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <font color="Maroon" title="c17">rh</font></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p4">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">rh</font></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:156_2"><span class="lab"><font color="Green" title="E74">C0</font></span></a>, <a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c12">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c14">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c17">rh</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span> <a href="funct_4.html#K1" title="FUNCT_4:func.1">+*</a> <span class="p1">(<span class="default"><span class="p2">(<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="c14">phii</font></span>)</span> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <span class="p3">(<span class="default"><a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> <a href="funcop_1.html#K16" title="FUNCOP_1:func.16">.--&gt;</a> <font color="Maroon" title="c18">u</font></span>)</span></span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">rh</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p3">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></span>
<br/>.= 
<font color="Maroon" title="c21">I2u</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c16">h</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_4.html#T71" title="FUNCT_4:th.71">FUNCT_4:71</a></span>
;<br/><a NAME="E7:156_2_1_2_2"/><span class="kw">then </span>
<font color="Maroon" title="c20">I1u</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c16">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156_2_1_2_2"><span class="lab"><font color="Green" title="E76">E0</font></span></a>, <a class="txt" href="#E1:156_2"><span class="lab"><font color="Green" title="E73">C1</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E8:156_2_1_2_2"/>
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:156_2_1_2"><span class="lab"><font color="Green" title="E75">D0</font></span></a>, <a class="ref" href="fomodel2.html#T19" title="FOMODEL2:th.19">FOMODEL2:19</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="E7:156_2_1_2"/><span class="kw">then </span>
( <font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 iff  not <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c14">phii</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T39" title="FOMODEL0:th.39">FOMODEL0:39</a></span>;<br/><span class="kw">hence </span><a NAME="E8:156_2_1_2"/>
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="fomodel0.html#T39" title="FOMODEL0:th.39">FOMODEL0:39</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><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="156_2_1_3"><span class="kw">suppose </span></a><a NAME="E1:156_2_1_3"/>
<font color="Maroon" title="c13">phi</font> is  <a href="fomodel1.html#V15" title="FOMODEL1:attr.15">0wff</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="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:156_2_1_3"/>
<font color="Maroon" title="c11">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c13">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:156"><span class="lab"><font color="Green" title="E71">B0</font></span></a>, <a class="txt" href="#E2:156_2"><span class="lab"><font color="Green" title="E74">C0</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><span class="kw">end;</span></div></div><span class="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E3:156"/><span class="lab"><font color="Green" title="E73">B2</font></span>: 
 for <font color="Olive" title="b1">n</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">n</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="#E1:156"><span class="lab"><font color="Green" title="E71">B0</font></span></a>, <a class="txt" href="#E2:156"><span class="lab"><font color="Green" title="E72">B1</font></span></a>);<br/></span>
<span class="kw">let </span><font color="Maroon" title="c10">I1</font>, <font color="Maroon" title="c11">I2</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">U</font> <a href="fomodel2.html#K16" title="FOMODEL2:func.16">-InterpretersOf</a> <font color="Maroon" title="c2">S</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="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> implies <font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font> )</span><br/>

<span class="kw">set </span><font color="Maroon" title="c12">d</font> =  <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font>;<br/>
<a NAME="E4:156"/>
<font color="Maroon" title="c3">phi</font> <a href="fomodel3.html#K25" title="FOMODEL3:func.25">null</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> is <span class="p1">(<span class="default"><a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a> 
 
;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">phii</font> = <font color="Maroon" title="c3">phi</font> as   <a href="fomodel2.html#K32" title="FOMODEL2:func.32">Depth</a> <font color="Maroon" title="c3">phi</font> <a href="fomodel2.html#V3" title="FOMODEL2:attr.3">-wff</a>  <a href="fomodel1.html#NM3" title="FOMODEL1:NM.3">string</a> of <font color="Maroon" title="c2">S</font> ;<br/>
<span class="kw">assume </span><a NAME="E6:156"/>
<font color="Maroon" title="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c3">phi</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></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="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font></span><br/>

<a NAME="E7:156"/><span class="kw">then </span>
<font color="Maroon" title="c10">I1</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="relat_1.html#K2" title="RELAT_1:func.2">rng</a> <font color="Maroon" title="c13">phii</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="fomodel1.html#K37" title="FOMODEL1:func.37">OwnSymbolsOf</a> <font color="Maroon" title="c2">S</font></span>)</span></span>)</span>
 
;<br/>
<span class="kw">hence </span><a NAME="E8:156"/>
<font color="Maroon" title="c10">I1</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">I2</font> <a href="fomodel2.html#K26" title="FOMODEL2:func.26">-TruthEval</a> <font color="Maroon" title="c3">phi</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:156"><span class="lab"><font color="Green" title="E73">B2</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>
