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

<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">F</font>, <font color="Olive" title="b2">G</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">F</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <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="Olive" title="b1">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="Olive" title="b2">G</font> holds <br/><font color="Maroon" title="c1">X</font> <a href="cqc_the1.html#R3" title="CQC_THE1:pred.3">|-</a> <font color="Olive" title="b1">F</font> <a href="cqc_lang.html#K9" title="CQC_LANG:func.9">=&gt;</a> <font color="Olive" title="b2">G</font></span><br/><span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">F</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a>  &amp; <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> implies <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> )</span><br/>





<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:111"/><span class="lab"><font color="Green" title="E72">A1</font></span>: 
<font color="Maroon" title="c2">F</font> is  <a href="qc_lang1.html#V6" title="QC_LANG1:attr.6">closed</a> 
 <span class="kw">and </span><br/><a NAME="E2:111"/><span class="lab"><font color="Green" title="E73">A2</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/>

<a NAME="E3:111"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111"><span class="lab"><font color="Green" title="E73">A2</font></span></a>, <a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E5:111"/><span class="lab"><font color="Green" title="E74">A3</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:111"/><span class="lab"><font color="Green" title="E75">A4</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T70" title="CQC_THE1:th.70">CQC_THE1:70</a></span>;<br/>
<a NAME="E7:111"/>
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="cqc_the1.html#D5" title="CQC_THE1:def.5">CQC_THE1:def 5</a></span>;<br/>
<a NAME="E8:111"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A5</font></span>: 
<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> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111"><span class="lab"><font color="Green" title="E75">A4</font></span></a>, <a class="ref" href="cqc_the1.html#D6" title="CQC_THE1:def.6">CQC_THE1:def 6</a></span>;<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>( 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="E9:111"/><span class="lab"><font color="Green" title="E77">A6</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 ( 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="111_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c5">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="Red">::  thesis: </font></span></a><span class="hide"> ( ( 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> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>] ) implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:111_1"/><span class="lab"><font color="Green" title="E78">A7</font></span>: 
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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">n</font>]</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E2:111_1"/><span class="lab"><font color="Green" title="E79">A8</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n</font>
 <span class="kw">and </span><br/><a NAME="E3:111_1"/><span class="lab"><font color="Green" title="E80">A9</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> 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> <font color="Maroon" title="c5">n</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="b1">H</font></span><br/>

<a NAME="E4:111_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> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></span>;<br/>
<a NAME="E5:111_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E81">A10</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="txt" href="#E2:111_1"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="ref" href="cqc_the1.html#D5" title="CQC_THE1:def.5">CQC_THE1:def 5</a></span>;<br/>
<span class="kw">let </span><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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="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>  implies <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> )</span><br/>

<span class="kw">assume </span><a NAME="E6:111_1"/><span class="lab"><font color="Green" title="E82">A11</font></span>: 
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_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="E1:111_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:111_1"><span class="lab"><font color="Green" title="E79">A8</font></span></a>, <a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="ref" href="cqc_the1.html#T45" title="CQC_THE1:th.45">CQC_THE1:45</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_1"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E83">A12</font></span>: 
( <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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="111_1_1_1_1_1"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_1_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="E1:111_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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1_1_1"><span class="lab"><font color="Green" title="E83">A12</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:111_1_1_1_1_1_1_1"/><span class="lab"><font color="Green" title="E84">A13</font></span>: 
<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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <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></span><br/><div class="add"><a NAME="E2:111_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T38" title="CQC_THE1:th.38">CQC_THE1:38</a></span>;<br/><a NAME="E3:111_1_1_1_1_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E85">A14</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E84">A13</font></span></a>, <a class="ref" href="cqc_the1.html#D9" title="CQC_THE1:def.9">CQC_THE1:def 9</a></span>;<br/><a NAME="E4:111_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="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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T45" title="LUKASI_1:th.45">LUKASI_1:45</a></span>;<br/><span class="kw">hence </span><a NAME="E5:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1_1_1_1_1_1_1"><span class="lab"><font color="Green" title="E85">A14</font></span></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E5:111_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_2"/><span class="kw">then </span>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><span class="kw">hence </span><a NAME="E3:111_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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a>, <a class="ref" href="lukasi_1.html#T50" title="LUKASI_1:th.50">LUKASI_1:50</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_3"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_3"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_3"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T100" title="CQC_THE1:th.100">CQC_THE1:100</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_4"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_4"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_4"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T101" title="CQC_THE1:th.101">CQC_THE1:101</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_5"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_5"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_5"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T102" title="CQC_THE1:th.102">CQC_THE1:102</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_6"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_6"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_6"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T103" title="CQC_THE1:th.103">CQC_THE1:103</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_7"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><a NAME="E2:111_1_1_1_7"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E3:111_1_1_1_7"/><span class="kw">then </span>
<font color="Maroon" title="c6">H</font> is  <a href="cqc_the1.html#V2" title="CQC_THE1:attr.2">valid</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the1.html#T105" title="CQC_THE1:th.105">CQC_THE1:105</a></span>;<br/><span class="kw">hence </span><a NAME="E4:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_8"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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> <span class="kw"> such that </span><br/><a NAME="E3:111_1_1_1_8"/><span class="lab"><font color="Green" title="E83">A15</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E4:111_1_1_1_8"/><span class="lab"><font color="Green" title="E84">A16</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:111_1_1_1_8"/><span class="lab"><font color="Green" title="E85">A17</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">j</font>
 <span class="kw">and </span><br/><a NAME="E6:111_1_1_1_8"/><span class="lab"><font color="Green" title="E86">A18</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E7:111_1_1_1_8"/><span class="lab"><font color="Green" title="E87">A19</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E8:111_1_1_1_8"/><span class="lab"><font color="Green" title="E88">A20</font></span>: 
<font color="Maroon" title="c10">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">H</font>
 <span class="kw">and </span><br/><a NAME="E9:111_1_1_1_8"/><span class="lab"><font color="Green" title="E89">A21</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E10:111_1_1_1_8"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="txt" href="#E4:111_1_1_1_8"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E11:111_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E3:111_1_1_1_8"><span class="lab"><font color="Green" title="E83">A15</font></span></a>, <a class="txt" href="#E4:111_1_1_1_8"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E9:111_1_1_1_8"><span class="lab"><font color="Green" title="E89">A21</font></span></a></span>;<br/><a NAME="E12:111_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>
 <span class="kw">by</span> <span class="lab"><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></span>;<br/><a NAME="E13:111_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E91">A23</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:111_1_1_1_8"><span class="lab"><font color="Green" title="E90">A22</font></span></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></span>;<br/><a NAME="E14:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_1_1_1_8"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E6:111_1_1_1_8"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E15:111_1_1_1_8"/><span class="kw">then </span><span class="lab"><font color="Green" title="E92">A24</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E16:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:111_1_1_1_8"><span class="lab"><font color="Green" title="E84">A16</font></span></a>, <a class="txt" href="#E6:111_1_1_1_8"><span class="lab"><font color="Green" title="E86">A18</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E17:111_1_1_1_8"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E5:111_1_1_1_8"><span class="lab"><font color="Green" title="E85">A17</font></span></a>, <a class="txt" href="#E7:111_1_1_1_8"><span class="lab"><font color="Green" title="E87">A19</font></span></a>, <a class="txt" href="#E15:111_1_1_1_8"><span class="lab"><font color="Green" title="E92">A24</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E18:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_1_1_1_8"><span class="lab"><font color="Green" title="E88">A20</font></span></a>, <a class="txt" href="#E13:111_1_1_1_8"><span class="lab"><font color="Green" title="E91">A23</font></span></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_9"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:111_1_1_1_9"/><span class="lab"><font color="Green" title="E83">A25</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E4:111_1_1_1_9"/><span class="lab"><font color="Green" title="E84">A26</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:111_1_1_1_9"/><span class="lab"><font color="Green" title="E85">A27</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E6:111_1_1_1_9"/><span class="lab"><font color="Green" title="E86">A28</font></span>: 
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>
 <span class="kw">and </span><br/><a NAME="E7:111_1_1_1_9"/><span class="lab"><font color="Green" title="E87">A29</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><a NAME="E8:111_1_1_1_9"/><span class="lab"><font color="Green" title="E88">A30</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1_1_1_9"><span class="lab"><font color="Green" title="E86">A28</font></span></a>, <a class="ref" href="cqc_the2.html#T78" target="_self" title="CQC_THE2:th.78">Th78</a>, <a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></span>;<br/><a NAME="E9:111_1_1_1_9"/>
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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a></span>;<br/><a NAME="E10:111_1_1_1_9"/><span class="kw">then </span><span class="lab"><font color="Green" title="E89">A31</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T78" target="_self" title="CQC_THE2:th.78">Th78</a>, <a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></span>;<br/><a NAME="E11:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="txt" href="#E4:111_1_1_1_9"><span class="lab"><font color="Green" title="E84">A26</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E12:111_1_1_1_9"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E3:111_1_1_1_9"><span class="lab"><font color="Green" title="E83">A25</font></span></a>, <a class="txt" href="#E4:111_1_1_1_9"><span class="lab"><font color="Green" title="E84">A26</font></span></a>, <a class="txt" href="#E5:111_1_1_1_9"><span class="lab"><font color="Green" title="E85">A27</font></span></a>, <a class="ref" href="cqc_the2.html#T94" target="_self" title="CQC_THE2:th.94">Th94</a></span>;<br/><a NAME="E13:111_1_1_1_9"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:111_1_1_1_9"><span class="lab"><font color="Green" title="E89">A31</font></span></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></span>;<br/><span class="kw">hence </span><a NAME="E14:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:111_1_1_1_9"><span class="lab"><font color="Green" title="E87">A29</font></span></a>, <a class="txt" href="#E8:111_1_1_1_9"><span class="lab"><font color="Green" title="E88">A30</font></span></a>, <a class="ref" href="lukasi_1.html#T76" title="LUKASI_1:th.76">LUKASI_1:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="111_1_1_1_10"><span class="kw">suppose </span></a><a NAME="E1:111_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
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="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></span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E3:111_1_1_1_10"/><span class="lab"><font color="Green" title="E83">A32</font></span>: 
1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">i</font>
 <span class="kw">and </span><br/><a NAME="E4:111_1_1_1_10"/><span class="lab"><font color="Green" title="E84">A33</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E5:111_1_1_1_10"/><span class="lab"><font color="Green" title="E85">A34</font></span>: 
( <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>  )
 <span class="kw">and </span><br/><a NAME="E6:111_1_1_1_10"/><span class="lab"><font color="Green" title="E86">A35</font></span>: 
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>
 <span class="kw">and </span><br/><a NAME="E7:111_1_1_1_10"/><span class="lab"><font color="Green" title="E87">A36</font></span>: 
<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> 
 <span class="kw">and </span><br/><a NAME="E8:111_1_1_1_10"/><span class="lab"><font color="Green" title="E88">A37</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1"><span class="lab"><font color="Green" title="E82">A11</font></span></a>, <a class="txt" href="#E5:111_1"><span class="lab"><font color="Green" title="E81">A10</font></span></a>, <a class="ref" href="cqc_the1.html#D4" title="CQC_THE1:def.4">CQC_THE1:def 4</a></span>;<br/><span class="kw">reconsider </span><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>  <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111_1_1_1_10"><span class="lab"><font color="Green" title="E85">A34</font></span></a></span>;<br/><a NAME="E10:111_1_1_1_10"/><span class="lab"><font color="Green" title="E89">A38</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:111_1_1_1_10"><span class="lab"><font color="Green" title="E86">A35</font></span></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></span>;<br/><a NAME="E11:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111"><span class="lab"><font color="Green" title="E72">A1</font></span></a>, <a class="ref" href="qc_lang1.html#D30" title="QC_LANG1:def.30">QC_LANG1:def 30</a></span>;<br/><a NAME="E12:111_1_1_1_10"/><span class="kw">then </span><span class="lab"><font color="Green" title="E90">A39</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the2.html#T78" target="_self" title="CQC_THE2:th.78">Th78</a>, <a class="ref" href="cqc_the1.html#T98" title="CQC_THE1:th.98">CQC_THE1:98</a></span>;<br/><a NAME="E13:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:111_1"><span class="lab"><font color="Green" title="E80">A9</font></span></a>, <a class="txt" href="#E4:111_1_1_1_10"><span class="lab"><font color="Green" title="E84">A33</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E14:111_1_1_1_10"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:111_1"><span class="lab"><font color="Green" title="E78">A7</font></span></a>, <a class="txt" href="#E3:111_1_1_1_10"><span class="lab"><font color="Green" title="E83">A32</font></span></a>, <a class="txt" href="#E4:111_1_1_1_10"><span class="lab"><font color="Green" title="E84">A33</font></span></a>, <a class="txt" href="#E7:111_1_1_1_10"><span class="lab"><font color="Green" title="E87">A36</font></span></a>, <a class="ref" href="cqc_the2.html#T94" target="_self" title="CQC_THE2:th.94">Th94</a></span>;<br/><a NAME="E15:111_1_1_1_10"/><span class="kw">then </span>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:111_1_1_1_10"><span class="lab"><font color="Green" title="E90">A39</font></span></a>, <a class="ref" href="cqc_the1.html#T92" title="CQC_THE1:th.92">CQC_THE1:92</a></span>;<br/><span class="kw">hence </span><a NAME="E16:111_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:111_1_1_1_10"><span class="lab"><font color="Green" title="E88">A37</font></span></a>, <a class="txt" href="#E10:111_1_1_1_10"><span class="lab"><font color="Green" title="E89">A38</font></span></a>, <a class="ref" href="lukasi_1.html#T76" title="LUKASI_1:th.76">LUKASI_1:76</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E8:111_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:111"/><span class="lab"><font color="Green" title="E78">A40</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#S4" title="NAT_1:sch.4">NAT_1:sch 4</a>(<a class="txt" href="#E9:111"><span class="lab"><font color="Green" title="E77">A6</font></span></a>);<br/></span>
<a NAME="E11:111"/>
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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:111"><span class="lab"><font color="Green" title="E74">A3</font></span></a>, <a class="ref" href="cqc_the1.html#T58" title="CQC_THE1:th.58">CQC_THE1:58</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:111"/>
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:111"><span class="lab"><font color="Green" title="E78">A40</font></span></a>, <a class="txt" href="#E8:111"><span class="lab"><font color="Green" title="E76">A5</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
