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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be    <a href="valuat_1.html#M1" target="_self">interpretation</a> of <font color="Maroon">c<sub>1</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="qc_lang1.html#NM8">QC-formula</a>;<br/>













<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>;<br/>
<div><i><font color="Green">E33</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>thus </b><a NAME="E1:47_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:47_1_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V2">atomic</a>
 ;<br/>

<b>thus </b><a NAME="E2:47_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E1:47_1_1_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:47_1_1_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>15</sub></font> being   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>14</sub></font>, <font color="Maroon">c<sub>16</sub></font> being    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">c<sub>14</sub></font><b> such that </b><br/><a NAME="E4:47_1_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">c<sub>16</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="qc_lang1.html#D17">QC_LANG1:def 17</a></i>;<br/>
<a NAME="E5:47_1_1_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_1_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E4:47_1_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_lang.html#T30">CQC_LANG:30</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> =  <a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<a NAME="E6:47_1_1_1"/><i><font color="Green">E39</font></i>: 
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T17">CQC_LANG:17</a></i>;<br/>
<a NAME="E7:47_1_1_1"/>
( <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K4">free_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( 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> <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> &amp; <span class="p1">(<span class="default"><a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="qc_lang1.html#K3">fixed_QC-variables</a>  ) </span>}</span>  <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T17">CQC_LANG:17</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> =  <a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>9</sub></font></span>)</span>, <font color="Maroon">c<sub>20</sub></font> =  <a href="cqc_lang.html#K5">Subst</a> <font color="Maroon">c<sub>16</sub></font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6">.--&gt;</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> as   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">c<sub>14</sub></font> <b>by </b><i><a class="txt" href="valuat_1.html#E6:47_1_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_lang.html#T15">CQC_LANG:15</a></i>;<br/>
<a NAME="E9:47_1_1_1"/><i><font color="Green">E40</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</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> <font color="Maroon">c<sub>14</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E10:47_1_1_1"/><i><font color="Green">E41</font></i>: 
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</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> <font color="Maroon">c<sub>14</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 
<b>by </b><i><a class="ref" href="valuat_1.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E11:47_1_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:47_1_1_1_1_1"/><i><font color="Green">E43</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> )
 ;<br/><a NAME="E2:47_1_1_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>21</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">c<sub>16</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="qc_lang1.html#D8">QC_LANG1:def 8</a></i>;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1_1_1_1"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang3.html#K3">a.</a> 0
 ;<br/><a NAME="E2:47_1_1_1_1_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_lang.html#T11">CQC_LANG:11</a></i>;<br/><a NAME="E3:47_1_1_1_1_1_1"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="valuat_1.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E4:47_1_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1_1_1_1"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1_1_1_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang3.html#K3">a.</a> 0
 ;<br/><a NAME="E2:47_1_1_1_1_1_2"/><b>then </b><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_lang.html#T11">CQC_LANG:11</a></i>;<br/><a NAME="E3:47_1_1_1_1_1_2"/>
( <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="valuat_1.html#D8" target="_self">Def8</a></i>;<br/><b>hence </b><a NAME="E4:47_1_1_1_1_1_2"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_1_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_1_1_1_1_2"><i><font color="Green">E46</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:47_1_1_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_1_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:47_1_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E9:47_1_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="valuat_1.html#E10:47_1_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="finseq_1.html#T18">FINSEQ_1:18</a></i>;<br/>


</div><b>end;</b></div>
<div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:47_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T16" target="_self">Th16</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E11:47_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="valuat_1.html#T16" target="_self">Th16</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_1_1_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_1_1_3"/><b>then </b>
not <font color="Maroon">c<sub>13</sub></font> <a href="valuat_1.html#K8" target="_self">*'</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="valuat_1.html#K11" target="_self">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T17" target="_self">Th17</a></i>;<br/><b>hence </b><a NAME="E3:47_1_1_1_3"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E11:47_1_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="valuat_1.html#T17" target="_self">Th17</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E14:47_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E12:47_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E2:47_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9">VERUM</a> ]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E1:47_1_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <a href="cqc_lang.html#K9">VERUM</a>  <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <a href="cqc_lang.html#K9">VERUM</a>  <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:47_1_2"/>
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E3:47_1_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9">VERUM</a>  )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T28">CQC_LANG:28</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/>


</div><b>end;</b></div><b>thus </b><a NAME="E3:47_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_3"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V3">negative</a>
 <b>and </b><br/><a NAME="E2:47_1_3"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:47_1_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E1:47_1_3_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:47_1_3_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E3:47_1_3_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_3"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T31">CQC_LANG:31</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</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="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="cqc_lang.html#T18">CQC_LANG:18</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <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="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T18">CQC_LANG:18</a></i>;<br/>
<div><i><font color="Green">E39</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:47_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E3:47_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E4:47_1_3_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_3_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E5:47_1_3_1_1"/><b>then </b>
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:47_1_3_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_3_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E3:47_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><a NAME="E4:47_1_3_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_3"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_3_1"><i><font color="Green">E37</font></i></a></i>;<br/><a NAME="E5:47_1_3_1_2"/><b>then </b>
 <a href="margrel1.html#K11">'not'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T41">MARGREL1:41</a></i>;<br/><b>hence </b><a NAME="E6:47_1_3_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_3_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="valuat_1.html#T20" target="_self">Th20</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:47_1_3_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:47_1_3_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E4:47_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_4"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a>
 <b>and </b><br/><a NAME="E2:47_1_4"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 <b>and </b><br/><a NAME="E3:47_1_4"/><i><font color="Green">E36</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 ;<br/>

<b>thus </b><a NAME="E4:47_1_4"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>10</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</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="E1:47_1_4_1"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>13</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:47_1_4_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/>

<a NAME="E3:47_1_4_1"/><i><font color="Green">E39</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T33">CQC_LANG:33</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>15</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>9</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="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="cqc_lang.html#T19">CQC_LANG:19</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <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="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="cqc_lang.html#T19">CQC_LANG:19</a></i>;<br/>
<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:47_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>;<br/><a NAME="E3:47_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><a NAME="E4:47_1_4_1_1"/><b>then </b>
( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  &amp; <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a>  )
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_4_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E5:47_1_4_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E6:47_1_4_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_2"/><i><font color="Green">E41</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_4_1_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_4"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_4_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:47_1_4_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:47_1_4_1_2_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_4_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_4_1_2_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_4"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_4_1"><i><font color="Green">E38</font></i></a></i>;<br/><a NAME="E3:47_1_4_1_2_2"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="margrel1.html#K12">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/><b>hence </b><a NAME="E4:47_1_4_1_2_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_4_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_4_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="valuat_1.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:47_1_4_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_4_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_4_1_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="margrel1.html#T45">MARGREL1:45</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E8:47_1_4_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:47_1_4_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div><b>thus </b><a NAME="E5:47_1"/>
( <font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>8</sub></font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5"><b>proof </b></a><div class="add">

<b>assume </b><b>that </b><br/><a NAME="E1:47_1_5"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>8</sub></font> is <a href="qc_lang1.html#V5">universal</a>
 <b>and </b><br/><a NAME="E2:47_1_5"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>9</sub></font> being   <a href="qc_lang1.html#NM3">bound_QC-variable</a>, <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="E4:47_1_5"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="qc_lang1.html#D20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E5:47_1_5"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="valuat_1.html#E4:47_1_5"><i><font color="Green">E36</font></i></a>, <a class="ref" href="qc_lang1.html#D26">QC_LANG1:def 26</a></i>;<br/>
<b>thus </b><a NAME="E6:47_1_5"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7">CQC-WFF</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/>for <font color="Olive">b<sub>5</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>3</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Olive">b<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>5</sub></font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>13</sub></font>, <font color="Maroon">c<sub>14</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="E1:47_1_5_1"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>15</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E2:47_1_5_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/>

<div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:47_1_5_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>11</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="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T23">CQC_LANG:23</a></i>;<br/><a NAME="E4:47_1_5_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a></i>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_1_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:47_1_5_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>12</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>17</sub></font> = <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>12</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="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T23">CQC_LANG:23</a></i>;<br/><a NAME="E4:47_1_5_1_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="cqc_lang.html#K15">All</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5"><i><font color="Green">E34</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_lang.html#T36">CQC_LANG:36</a></i>;<br/><a NAME="E5:47_1_5_1_1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>16</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> &amp;  <a href="valuat_1.html#K12" target="_self">Valid</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<font color="Maroon">c<sub>17</sub></font></span>)</span>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#R2">=</a>  <a href="valuat_1.html#K7" target="_self">FOR_ALL</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="valuat_1.html#E20"><i><font color="Green">Lemma13</font></i></a></i>;<br/><div><i><font color="Green">E47</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_1_1_1"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 ;<br/><a NAME="E2:47_1_5_1_1_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st ( for <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> ) holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>18</sub></font> be    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E1:47_1_5_1_1_1_1_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>

<a NAME="E2:47_1_5_1_1_1_1_1"/><b>then </b><i><font color="Green">E50</font></i>: 
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E3:47_1_5_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 
<b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="valuat_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E4:47_1_5_1_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_5"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_5_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_5_1_1_1_1_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_1_5_1_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K8">TRUE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="valuat_1.html#T8" target="_self">Th8</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_1_1_2"/><i><font color="Green">E48</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 ;<br/><a NAME="E2:47_1_5_1_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_1_2_1"><b>proof </b></a><div class="add">

<b>consider </b><font color="Maroon">c<sub>18</sub></font> being    <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E2:47_1_5_1_1_1_2_1"/><i><font color="Green">E49</font></i>: 
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>and </b><br/><a NAME="E3:47_1_5_1_1_1_2_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="valuat_1.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E4:47_1_5_1_1_1_2_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>12</sub></font> )
 
<b>by </b><i><a class="txt" href="valuat_1.html#E1:47_1_5_1_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="valuat_1.html#E3:47_1_5_1_1_1_2_1"><i><font color="Green">E50</font></i></a></i>;<br/>
<a NAME="E5:47_1_5_1_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 
<b>by </b><i><a class="txt" href="valuat_1.html#E2:47_1_5"><i><font color="Green">E35</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_5_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="valuat_1.html#E2:47_1_5_1_1_1_2_1"><i><font color="Green">E49</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:47_1_5_1_1_1_2_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <br/>( <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>17</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a>  &amp; ( for <font color="Olive">b<sub>2</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a> st  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> ) )
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_5_1_1_1_2_1"><i><font color="Green">E50</font></i></a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E3:47_1_5_1_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="margrel1.html#K7">FALSE</a> 
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5_1_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="valuat_1.html#T7" target="_self">Th7</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:47_1_5_1_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E6:47_1_5_1_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="margrel1.html#T39">MARGREL1:39</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_1_2"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:47_1_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T37">CQC_LANG:37</a></i>;<br/><b>hence </b><a NAME="E3:47_1_5_1_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="cqc_lang.html#T40">CQC_LANG:40</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:47_1_5_1_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E5:47_1_5_1_1"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="47_1_5_1_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:47_1_5_1_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K20">bound_in</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><a NAME="E2:47_1_5_1_2"/><b>then </b>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K14">All</a> <font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="cqc_lang.html#T37">CQC_LANG:37</a></i>;<br/><b>hence </b><a NAME="E3:47_1_5_1_2"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E4:47_1_5"><i><font color="Green">E36</font></i></a>, <a class="txt" href="valuat_1.html#E5:47_1_5"><i><font color="Green">E37</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="valuat_1.html#E1:47_1_5_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="cqc_lang.html#T40">CQC_LANG:40</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:47_1_5_1"/>
<span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>13</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="valuat_1.html#E3:47_1_5_1"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>


</div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E2:47"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</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="qc_lang1.html#S2">QC_LANG1:sch 2</a>(<a class="txt" href="valuat_1.html#E1:47"><i><font color="Green">E33</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:47"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="cqc_lang.html#K17">.</a> <font color="Maroon">c<sub>3</sub></font> implies for <font color="Olive">b<sub>1</sub></font> being   <a href="fraenkel.html#M2">Element</a> of  <a href="valuat_1.html#K2" target="_self">Valuations_in</a> <font color="Maroon">c<sub>1</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>3</sub></font> holds <br/><span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="valuat_1.html#K12" target="_self">Valid</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>6</sub></font></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> )
 ;<br/>


</div>
