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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  <a href="henmodel.html#V1">Consistent</a> <a href="subset_1.html#NM2">Subset</a> of <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="henmodel.html#M1">Henkin_interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>





<b>assume </b><a NAME="E1:15"/><i><font color="Green">E18</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V1" target="_self">negation_faithful</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V2" target="_self">with_examples</a> holds <br/>( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/>

<b>assume </b><a NAME="E2:15"/><i><font color="Green">E19</font></i>: 
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V1" target="_self">negation_faithful</a> &amp; <font color="Maroon">c<sub>1</sub></font> is <a href="goedelcp.html#V2" target="_self">with_examples</a> )
 ;<br/>

<a NAME="E3:15"/><i><font color="Green">E20</font></i>: 
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> implies ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>4</sub></font> ) )
 
<b>by </b><i><a class="txt" href="goedelcp.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E21</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E2:15_1"/><b>then </b>
1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E4:15_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i><a class="ref" href="substut2.html#T34">SUBSTUT2:34</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being    <a href="substut2.html#M1">PATH</a> of <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>;<br/><a NAME="E6:15_1"/><i><font color="Green">E23</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> holds <br/>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  st <br/>( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Olive">b<sub>3</sub></font> ) ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="substut2.html#D6">SUBSTUT2:def 6</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> implies ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) ) );<br/><a NAME="E7:15_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E8:15_1"/><i><font color="Green">E25</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:15_1_1"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/>

<b>assume </b><a NAME="E2:15_1_1"/><i><font color="Green">E27</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>
<div><i><font color="Green">E28</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:15_1_1_1"/><b>then </b><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="substut2.html#D6">SUBSTUT2:def 6</a></i>;<br/><b>take </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E3:15_1_1_1"/>
( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>9</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="goedelcp.html#T15" target="_self">Th15</a></i>;<br/><b>hence </b><a NAME="E4:15_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E2:15_1_1_2"/><b>then </b>
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 <b>by </b><i><a class="ref" href="nat_1.html#T19">NAT_1:19</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:15_1_1_2"/><b>then </b><i><font color="Green">E29</font></i>: 
( 0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E5:15_1_1_2"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="qc_lang2.html#R1">is_immediate_constituent_of</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="ref" href="substut2.html#D6">SUBSTUT2:def 6</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E7:15_1_1_2"/><i><font color="Green">E31</font></i>: 
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>11</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>11</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>11</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>11</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="goedelcp.html#E3:15_1_1_2"><i><font color="Green">E29</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>12</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a><b> such that </b><br/><a NAME="E9:15_1_1_2"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="qc_lang2.html#R2">is_subformula_of</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goedelcp.html#E2:15_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="substut2.html#T27">SUBSTUT2:27</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>10</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goedelcp.html#E2:15_1_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="substut2.html#T28">SUBSTUT2:28</a></i>;<br/><b>take </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font>;<br/><div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:15_1_1_2_1"/><b>then </b>
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="goedelcp.html#T4" target="_self">Th4</a>, <a class="ref" href="cqc_sim1.html#T16">CQC_SIM1:16</a></i>;<br/><b>hence </b><a NAME="E3:15_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 ;<br/></div><b>end;</b></div><a NAME="E12:15_1_1_2"/><i><font color="Green">E34</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1"><i><font color="Green">E21</font></i></a>, <a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E9:15_1_1_2"><i><font color="Green">E32</font></i></a>, <a class="ref" href="substut2.html#T30">SUBSTUT2:30</a></i>;<br/><div><i><font color="Green">E35</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_2"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>15</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a><b> such that </b><a NAME="E1:15_1_1_2_2"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_lang.html#T19">CQC_LANG:19</a></i>;<br/><a NAME="E3:15_1_1_2_2"/><i><font color="Green">E37</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>;<br/><a NAME="E4:15_1_1_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:15_1_1_2_2"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goedelcp.html#E12:15_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:15_1_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:15_1_1_2_2"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>16</sub></font> iff <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E8:15_1_1_2_2"/><b>then </b><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goedelcp.html#E1:15_1_1_2_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="goedelcp.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E9:15_1_1_2_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:15_1_1_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:15_1_1_2_2"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:15_1_1_2_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E8:15_1_1_2_2"><i><font color="Green">E38</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_3"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>15</sub></font> being   <a href="qc_lang1.html#NM8">QC-formula</a><b> such that </b><a NAME="E1:15_1_1_2_3"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>15</sub></font> as    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_lang.html#T19">CQC_LANG:19</a></i>;<br/><a NAME="E3:15_1_1_2_3"/><i><font color="Green">E38</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>;<br/><a NAME="E4:15_1_1_2_3"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E4:15_1"><i><font color="Green">E22</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E5:15_1_1_2_3"/><b>then </b>
1 <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goedelcp.html#E12:15_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:15_1_1_2_3"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E7:15_1_1_2_3"/><b>then </b>
( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>16</sub></font> iff <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a></i>;<br/><a NAME="E8:15_1_1_2_3"/><b>then </b><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goedelcp.html#E1:15_1_1_2_3"><i><font color="Green">E37</font></i></a>, <a class="ref" href="goedelcp.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E9:15_1_1_2_3"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:15_1_1_2_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><a NAME="E10:15_1_1_2_3"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E11:15_1_1_2_3"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E8:15_1_1_2_3"><i><font color="Green">E39</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4"><b>now </b></a><div class="add"><b>given </b><font color="Maroon">c<sub>15</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><a NAME="E1:15_1_1_2_4"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K15">All</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>11</sub></font>
 ;<br/><a NAME="E2:15_1_1_2_4"/><i><font color="Green">E38</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15_1_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_sim1.html#T18">CQC_SIM1:18</a></i>;<br/><a NAME="E3:15_1_1_2_4"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="goedelcp.html#E12:15_1_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:15_1_1_2_4"/><b>then </b>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E5:15_1_1_2_4"/><b>then </b><i><font color="Green">E39</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T16">CQC_SIM1:16</a></i>;<br/><a NAME="E6:15_1_1_2_4"/><i><font color="Green">E40</font></i>: 
 <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goedelcp.html#E2:15_1_1_2_4"><i><font color="Green">E38</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_4_1"/>
<font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:15_1_1_2_4_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="ref" href="goedelcp.html#T10" target="_self">Th10</a></i>;<br/><a NAME="E4:15_1_1_2_4_1"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E5:15_1_1_2_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goedelcp.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E5:15_1_1_2_4_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E3:15_1_1_2_4_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E6:15_1_1_2_4_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="ref" href="goedelcp.html#T3" target="_self">Th3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1_1_2_4_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:15_1_1_2_4_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>16</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a><b> such that </b><br/><a NAME="E3:15_1_1_2_4_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="ref" href="goedelcp.html#T3" target="_self">Th3</a></i>;<br/><a NAME="E4:15_1_1_2_4_2"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E5:15_1_1_2_4"><i><font color="Green">E39</font></i></a>, <a class="ref" href="goedelcp.html#T14" target="_self">Th14</a></i>;<br/><a NAME="E5:15_1_1_2_4_2"/><b>then </b>
<font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span> <a href="substut2.html#K4">.</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="goedelcp.html#E1:15"><i><font color="Green">E18</font></i></a>, <a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E3:15_1_1_2_4_2"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E6:15_1_1_2_4_2"/>
<font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p1">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="goedelcp.html#T10" target="_self">Th10</a></i>;<br/></div><b>end;</b></div><a NAME="E9:15_1_1_2_4"/><b>then </b>
( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a>  <a href="cqc_lang.html#K10">'not'</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K16">Ex</a> <font color="Maroon">c<sub>15</sub></font>,<span class="p2">(<span class="default"><a href="cqc_lang.html#K10">'not'</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2_4"><i><font color="Green">E41</font></i></a>, <a class="ref" href="goedelcp.html#T1" target="_self">Th1</a>, <a class="ref" href="valuat_1.html#T28">VALUAT_1:28</a></i>;<br/><a NAME="E10:15_1_1_2_4"/><b>then </b>
( <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> ) &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>14</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>14</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E1:15_1_1_2_4"><i><font color="Green">E37</font></i></a>, <a class="ref" href="goedelcp.html#T11" target="_self">Th11</a>, <a class="ref" href="goedelcp.html#T12" target="_self">Th12</a></i>;<br/><b>hence </b><a NAME="E11:15_1_1_2_4"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E6:15_1_1_2_4"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:15_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E5:15_1_1_2"><i><font color="Green">E30</font></i></a>, <a class="txt" href="goedelcp.html#E7:15_1_1_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="goedelcp.html#E11:15_1_1_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="goedelcp.html#E13:15_1_1_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="goedelcp.html#E14:15_1_1_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="qc_lang2.html#D20">QC_LANG2:def 20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Olive">b<sub>1</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Olive">b<sub>1</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Olive">b<sub>1</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E3:15_1_1"><i><font color="Green">E28</font></i></a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:15_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="goedelcp.html#E7:15_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="goedelcp.html#E8:15_1"><i><font color="Green">E25</font></i></a>);<br/></i><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> <b> such that </b><br/><a NAME="E11:15_1"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>7</sub></font> &amp; ( <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> ) &amp; ( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>7</sub></font> implies <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>7</sub></font> ) )
 <b>by </b><i><a class="txt" href="goedelcp.html#E6:15_1"><i><font color="Green">E23</font></i></a></i>;<br/><b>thus </b><a NAME="E12:15_1"/>
( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E6:15_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="goedelcp.html#E11:15_1"><i><font color="Green">E26</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:15"/>
( <font color="Maroon">c<sub>2</sub></font>, <a href="henmodel.html#K4">valH</a>  <a href="valuat_1.html#R1">|=</a> <font color="Maroon">c<sub>4</sub></font> iff <font color="Maroon">c<sub>1</sub></font> <a href="henmodel.html#R1">|-</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="goedelcp.html#E2:15"><i><font color="Green">E19</font></i></a>, <a class="txt" href="goedelcp.html#E3:15"><i><font color="Green">E20</font></i></a>, <a class="ref" href="nat_1.html#T26">NAT_1:26</a></i>;<br/>


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