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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</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>3</sub></font>, <font color="Maroon">c<sub>4</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/>







<b>set </b><font color="Maroon">c<sub>5</sub></font> = <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<a NAME="E1:27"/><i><font color="Green">E17</font></i>: 
<span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="sublemma.html#K7">CQCSub_&amp;</a> <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>,<span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>
 
<b>by </b><i><a class="ref" href="substut2.html#T19" target="_self">Th19</a></i>;<br/>
<a NAME="E2:27"/><i><font color="Green">E18</font></i>: 
( <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="substut2.html#K5" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="substut2.html#K5" target="_self">`2</a>  <a href="hidden.html#R1">=</a>  <a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E3:27"/><b>then </b><i><font color="Green">E19</font></i>: 
<span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K21">Sub_&amp;</a> <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>,<span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="substut2.html#E1:27"><i><font color="Green">E17</font></i></a>, <a class="ref" href="sublemma.html#D4">SUBLEMMA:def 4</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
 <a href="substut1.html#K39">CQC_Sub</a> <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span>

<br/>.= 
<span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<span class="p3">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E2:27"><i><font color="Green">E18</font></i></a>, <a class="txt" href="substut2.html#E3:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="substut1.html#T31">SUBSTUT1:31</a></i>
;<br/>
<b>then </b><i><font color="Green">E20</font></i>: <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="substut2.html#K3" target="_self">Sbst</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>

<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>

;<br/>
<a NAME="E6:27"/>
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> implies  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:27_3"/><i><font color="Green">E21</font></i>: 
(  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> )
 ;<br/>

<a NAME="E2:27_3"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut2.html#E5:27"><i><font color="Green">E20</font></i></a>, <a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>;<br/>
<b>hence </b><a NAME="E3:27_3"/>
 <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="substut2.html#E1:27_3"><i><font color="Green">E21</font></i></a>, <a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:27"/>
( <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="cqc_lang.html#K11">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp; (  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> &amp;  <a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> implies  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="substut2.html#K4" target="_self">.</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> ) )
 <b>by </b><i><a class="txt" href="substut2.html#E5:27"><i><font color="Green">E20</font></i></a></i>;<br/>


</div>
