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

<b>let </b><font color="Maroon" title="c1">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon" title="c2">F</font>, <font color="Maroon" title="c3">G</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:108"/><i><font color="Green" title="E73">A1</font></i>: 
<font color="Maroon" title="c2">F</font> is <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>
 <b>and </b><br/><a NAME="E2:108"/><i><font color="Green" title="E74">A2</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c3">G</font>
 ;<br/>

<a NAME="E3:108"/>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E2:108"><i><font color="Green" title="E74">A2</font></i></a>, <a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c4">f</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">[:</a><span class="default"><a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ,<a href="cqc_the1.html#K2" title="CQC_THE1:func.2">Proof_Step_Kinds</a> </span><a href="zfmisc_1.html#K2" title="ZFMISC_1:func.2">:]</a></span><b> such that </b><br/><a NAME="E5:108"/><i><font color="Green" title="E75">A3</font></i>: 
<font color="Maroon" title="c4">f</font> <a href="cqc_the1.html#R2" title="CQC_THE1:pred.2">is_a_proof_wrt</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>and </b><br/><a NAME="E6:108"/><i><font color="Green" title="E76">A4</font></i>: 
 <a href="cqc_the1.html#K3" title="CQC_THE1:func.3">Effect</a> <font color="Maroon" title="c4">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T70" title="CQC_THE1:th.70">CQC_THE1:70</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b>( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> $1 &amp; $1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> implies for <font color="Olive" title="b1">H</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b1">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> $1</span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b1">H</font> );<br/>
<a NAME="E7:108"/><i><font color="Green" title="E77">A5</font></i>: 
for <font color="Olive" title="b1">n</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st ( for <font color="Olive" title="b2">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">n</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">k</font>] ) holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">n</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:108_1"/><i><font color="Green" title="E78">A6</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">k</font> &amp; <font color="Olive" title="b1">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> holds <br/>for <font color="Olive" title="b2">H</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>   st <font color="Olive" title="b2">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  holds <br/><font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b2">H</font>
 ;<br/>

<b>assume </b><a NAME="E2:108_1"/><i><font color="Green" title="E79">A7</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 ;<br/>

<b>let </b><font color="Maroon" title="c6">H</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E3:108_1"/><i><font color="Green" title="E80">A8</font></i>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 ;<br/>

<a NAME="E4:108_1"/>
<font color="Maroon" title="c5">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<a NAME="E5:108_1"/><b>then </b><i><font color="Green" title="E81">A9</font></i>: 
<font color="Maroon" title="c4">f</font>,<font color="Maroon" title="c5">n</font> <a href="cqc_the1.html#R1" title="CQC_THE1:pred.1">is_a_correct_step_wrt</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E5:108"><i><font color="Green" title="E75">A3</font></i></a>, <a class="txt" href="cqc_the2.html#E2:108_1"><i><font color="Green" title="E79">A7</font></i></a>, <a class="ref" href="cqc_the1.html#D5" title="CQC_THE1:def.5">CQC_THE1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8 or <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9 )
 </span><b>by </b><i><a class="txt" href="cqc_the2.html#E2:108_1"><i><font color="Green" title="E79">A7</font></i></a>, <a class="ref" href="cqc_the1.html#T45" title="CQC_THE1:th.45">CQC_THE1:45</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_1"/><b>then </b>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_1"/><b>then </b><i><font color="Green" title="E82">A10</font></i>: 
( <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c2">F</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="108_1_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:108_1_1_1_1_1_1"/>
( <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> or <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font> )
 </span><b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1_1_1_1"><i><font color="Green" title="E82">A10</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:108_1_1_1_1_1_1_1"/><i><font color="Green" title="E83">A11</font></i>: 
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T38" title="CQC_THE1:th.38">CQC_THE1:38</a></i>;<br/><a NAME="E3:108_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c6">H</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_the1.html#K1" title="CQC_THE1:func.1">Cn</a> <font color="Maroon" title="c1">X</font> &amp;  <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108_1_1_1_1_1_1_1"><i><font color="Green" title="E83">A11</font></i></a>, <a class="ref" href="lukasi_1.html#T45" title="LUKASI_1:th.45">LUKASI_1:45</a></i>;<br/><a NAME="E4:108_1_1_1_1_1_1_1"/><b>then </b>
( <font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c6">H</font> &amp; <font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c6">H</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font></span>)</span> )
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></i>;<br/><b>hence </b><a NAME="E5:108_1_1_1_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">F</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:108_1_1_1_1_1_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T44" title="LUKASI_1:th.44">LUKASI_1:44</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:108_1_1_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_2"><b>suppose </b></a><a NAME="E1:108_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_2"/><b>then </b>
<font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_2"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="lukasi_1.html#T50" title="LUKASI_1:th.50">LUKASI_1:50</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_2"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_3"><b>suppose </b></a><a NAME="E1:108_1_1_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 2
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_3"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_3"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T100" title="CQC_THE1:th.100">CQC_THE1:100</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_3"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T78" title="LUKASI_1:th.78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_4"><b>suppose </b></a><a NAME="E1:108_1_1_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 3
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_4"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_4"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T101" title="CQC_THE1:th.101">CQC_THE1:101</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_4"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T78" title="LUKASI_1:th.78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_5"><b>suppose </b></a><a NAME="E1:108_1_1_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 4
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_5"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font>, <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p3">(<span class="default"><font color="Olive" title="b2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b3">r</font></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K7" title="CQC_LANG:func.7">'not'</a> <span class="p3">(<span class="default"><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b3">r</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_5"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T102" title="CQC_THE1:th.102">CQC_THE1:102</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_5"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T78" title="LUKASI_1:th.78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_6"><b>suppose </b></a><a NAME="E1:108_1_1_1_6"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 5
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_6"/><b>then </b>
 ex <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  st <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">p</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b2">q</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">q</font> <a href="cqc_lang.html#K8" title="CQC_LANG:func.8">'&amp;'</a> <font color="Olive" title="b1">p</font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_6"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T103" title="CQC_THE1:th.103">CQC_THE1:103</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_6"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T78" title="LUKASI_1:th.78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_7"><b>suppose </b></a><a NAME="E1:108_1_1_1_7"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 6
 ;<br/><div class="add"><a NAME="E2:108_1_1_1_7"/><b>then </b>
 ex <font color="Olive" title="b1">p</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  ex <font color="Olive" title="b2">x</font> being  <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a> st <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Olive" title="b2">x</font>,<font color="Olive" title="b1">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b1">p</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E3:108_1_1_1_7"/><b>then </b>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T105" title="CQC_THE1:th.105">CQC_THE1:105</a></i>;<br/><b>hence </b><a NAME="E4:108_1_1_1_7"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T78" title="LUKASI_1:th.78">LUKASI_1:78</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_8"><b>suppose </b></a><a NAME="E1:108_1_1_1_8"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 7
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">i</font>, <font color="Maroon" title="c8">j</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c9">p</font>, <font color="Maroon" title="c10">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> <b> such that </b><br/><a NAME="E3:108_1_1_1_8"/><i><font color="Green" title="E82">A12</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">i</font> )
 <b>and </b><br/><a NAME="E4:108_1_1_1_8"/><i><font color="Green" title="E83">A13</font></i>: 
( <font color="Maroon" title="c9">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">j</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c10">q</font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E5:108_1_1_1_8"/>
( <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1_1_1_8"><i><font color="Green" title="E82">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E6:108_1_1_1_8"/><b>then </b>
( <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> &amp; 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font> &amp; <font color="Maroon" title="c8">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:108_1"><i><font color="Green" title="E79">A7</font></i></a>, <a class="txt" href="cqc_the2.html#E3:108_1_1_1_8"><i><font color="Green" title="E82">A12</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E7:108_1_1_1_8"/><b>then </b><i><font color="Green" title="E84">A14</font></i>: 
( <font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c10">q</font></span>)</span> &amp; <font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">p</font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="cqc_the2.html#E4:108_1_1_1_8"><i><font color="Green" title="E83">A13</font></i></a></i>;<br/><a NAME="E8:108_1_1_1_8"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c9">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c10">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T54" title="LUKASI_1:th.54">LUKASI_1:54</a></i>;<br/><a NAME="E9:108_1_1_1_8"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">p</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c10">q</font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E7:108_1_1_1_8"><i><font color="Green" title="E84">A14</font></i></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></i>;<br/><b>hence </b><a NAME="E10:108_1_1_1_8"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E4:108_1_1_1_8"><i><font color="Green" title="E83">A13</font></i></a>, <a class="txt" href="cqc_the2.html#E7:108_1_1_1_8"><i><font color="Green" title="E84">A14</font></i></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_9"><b>suppose </b></a><a NAME="E1:108_1_1_1_9"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 8
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c8">p</font>, <font color="Maroon" title="c9">q</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a> , <font color="Maroon" title="c10">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:108_1_1_1_9"/><i><font color="Green" title="E82">A15</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> )
 <b>and </b><br/><a NAME="E4:108_1_1_1_9"/><i><font color="Green" title="E83">A16</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font>
 <b>and </b><br/><a NAME="E5:108_1_1_1_9"/><i><font color="Green" title="E84">A17</font></i>: 
( not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c8">p</font> &amp; <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">q</font></span>)</span> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><a NAME="E6:108_1_1_1_9"/><i><font color="Green" title="E85">A18</font></i>: 
not <font color="Maroon" title="c10">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a></i>;<br/><a NAME="E7:108_1_1_1_9"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:108_1"><i><font color="Green" title="E79">A7</font></i></a>, <a class="txt" href="cqc_the2.html#E3:108_1_1_1_9"><i><font color="Green" title="E82">A15</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:108_1_1_1_9"/><b>then </b><i><font color="Green" title="E86">A19</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="cqc_the2.html#E3:108_1_1_1_9"><i><font color="Green" title="E82">A15</font></i></a>, <a class="txt" href="cqc_the2.html#E4:108_1_1_1_9"><i><font color="Green" title="E83">A16</font></i></a>, <a class="ref" href="cqc_the2.html#T94" target="_self" title="CQC_THE2:th.94">Th94</a></i>;<br/><a NAME="E9:108_1_1_1_9"/>
 <a href="cqc_the1.html#NR5" title="CQC_THE1:NR.5">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E6:108_1_1_1_9"><i><font color="Green" title="E85">A18</font></i></a>, <a class="ref" href="cqc_the2.html#T78" target="_self" title="CQC_THE2:th.78">Th78</a></i>;<br/><a NAME="E10:108_1_1_1_9"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p3">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></i>;<br/><a NAME="E11:108_1_1_1_9"/><b>then </b><i><font color="Green" title="E87">A20</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E8:108_1_1_1_9"><i><font color="Green" title="E86">A19</font></i></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></i>;<br/><a NAME="E12:108_1_1_1_9"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c9">q</font></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">p</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c10">x</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:108_1_1_1_9"><i><font color="Green" title="E84">A17</font></i></a>, <a class="ref" href="cqc_the2.html#T95" target="_self" title="CQC_THE2:th.95">Th95</a></i>;<br/><b>hence </b><a NAME="E13:108_1_1_1_9"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:108_1_1_1_9"><i><font color="Green" title="E84">A17</font></i></a>, <a class="txt" href="cqc_the2.html#E11:108_1_1_1_9"><i><font color="Green" title="E87">A20</font></i></a>, <a class="ref" href="lukasi_1.html#T76" title="LUKASI_1:th.76">LUKASI_1:76</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="108_1_1_1_10"><b>suppose </b></a><a NAME="E1:108_1_1_1_10"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="mcart_1.html#K2" title="MCART_1:func.2">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 9
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c7">i</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> , <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>, <font color="Maroon" title="c10">s</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><b> such that </b><br/><a NAME="E3:108_1_1_1_10"/><i><font color="Green" title="E82">A21</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font> &amp; <font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c5">n</font> )
 <b>and </b><br/><a NAME="E4:108_1_1_1_10"/><i><font color="Green" title="E83">A22</font></i>: 
( <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  &amp; <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  )
 <b>and </b><br/><a NAME="E5:108_1_1_1_10"/><i><font color="Green" title="E84">A23</font></i>: 
( not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c10">s</font> &amp; <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c8">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">i</font></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a>  &amp; <font color="Maroon" title="c6">H</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">y</font> )
 <b>by </b><i><a class="txt" href="cqc_the2.html#E3:108_1"><i><font color="Green" title="E80">A8</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1"><i><font color="Green" title="E81">A9</font></i></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">s1</font> = <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c12">s2</font> = <font color="Maroon" title="c10">s</font> <a href="cqc_lang.html#K14" title="CQC_LANG:func.14">.</a> <font color="Maroon" title="c9">y</font> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="cqc_lang.html#K4" title="CQC_LANG:func.4">CQC-WFF</a>  <b>by </b><i><a class="txt" href="cqc_the2.html#E4:108_1_1_1_10"><i><font color="Green" title="E83">A22</font></i></a></i>;<br/><a NAME="E7:108_1_1_1_10"/>
<font color="Maroon" title="c7">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E2:108_1"><i><font color="Green" title="E79">A7</font></i></a>, <a class="txt" href="cqc_the2.html#E3:108_1_1_1_10"><i><font color="Green" title="E82">A21</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E8:108_1_1_1_10"/><b>then </b><i><font color="Green" title="E85">A24</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a>  <a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c8">x</font>,<span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c11">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108_1"><i><font color="Green" title="E78">A6</font></i></a>, <a class="txt" href="cqc_the2.html#E3:108_1_1_1_10"><i><font color="Green" title="E82">A21</font></i></a>, <a class="txt" href="cqc_the2.html#E5:108_1_1_1_10"><i><font color="Green" title="E84">A23</font></i></a>, <a class="ref" href="cqc_the2.html#T94" target="_self" title="CQC_THE2:th.94">Th94</a></i>;<br/><a NAME="E9:108_1_1_1_10"/>
not <font color="Maroon" title="c8">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">F</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E1:108"><i><font color="Green" title="E73">A1</font></i></a>, <a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a></i>;<br/><a NAME="E10:108_1_1_1_10"/><b>then </b>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c8">x</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c11">s1</font></span>)</span></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">s1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_the2.html#T95" target="_self" title="CQC_THE2:th.95">Th95</a></i>;<br/><a NAME="E11:108_1_1_1_10"/><b>then </b><i><font color="Green" title="E86">A25</font></i>: 
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">s1</font></span>)</span>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E8:108_1_1_1_10"><i><font color="Green" title="E85">A24</font></i></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></i>;<br/><a NAME="E12:108_1_1_1_10"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K12" title="CQC_LANG:func.12">All</a> <font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c11">s1</font></span>)</span> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c12">s2</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:108_1_1_1_10"><i><font color="Green" title="E84">A23</font></i></a>, <a class="ref" href="cqc_the2.html#T28" target="_self" title="CQC_THE2:th.28">Th28</a>, <a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></i>;<br/><b>hence </b><a NAME="E13:108_1_1_1_10"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E5:108_1_1_1_10"><i><font color="Green" title="E84">A23</font></i></a>, <a class="txt" href="cqc_the2.html#E11:108_1_1_1_10"><i><font color="Green" title="E86">A25</font></i></a>, <a class="ref" href="lukasi_1.html#T76" title="LUKASI_1:th.76">LUKASI_1:76</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E7:108_1"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c6">H</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E8:108"/><i><font color="Green" title="E78">A26</font></i>: 
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>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="cqc_the2.html#E7:108"><i><font color="Green" title="E77">A5</font></i></a>);<br/></i>
<a NAME="E9:108"/><i><font color="Green" title="E79">A27</font></i>: 
( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> &amp;  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font> )
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E5:108"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="cqc_the1.html#T58" title="CQC_THE1:th.58">CQC_THE1:58</a></i>;<br/>
<a NAME="E10:108"/>
<font color="Maroon" title="c4">f</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E5:108"><i><font color="Green" title="E75">A3</font></i></a>, <a class="ref" href="cqc_the1.html#D5" title="CQC_THE1:def.5">CQC_THE1:def 5</a></i>;<br/>
<a NAME="E11:108"/><b>then </b>
<font color="Maroon" title="c3">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c4">f</font></span>)</span></span>)</span> <a href="mcart_1.html#K1" title="MCART_1:func.1">`1</a> 
 
<b>by </b><i><a class="txt" href="cqc_the2.html#E6:108"><i><font color="Green" title="E76">A4</font></i></a>, <a class="ref" href="cqc_the1.html#D6" title="CQC_THE1:def.6">CQC_THE1:def 6</a></i>;<br/>
<b>hence </b><a NAME="E12:108"/>
<font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Maroon" title="c2">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Maroon" title="c3">G</font>
 <b>by </b><i><a class="txt" href="cqc_the2.html#E8:108"><i><font color="Green" title="E78">A26</font></i></a>, <a class="txt" href="cqc_the2.html#E9:108"><i><font color="Green" title="E79">A27</font></i></a></i>;<br/>


</div>
