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<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>assume </b><a NAME="E1:28"/><i><font color="Green">E18</font></i>: 
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a> holds  <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"><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>,<font color="Olive">b<sub>1</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="substut1.html#NM1">CQC_Substitution</a> holds  <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"><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>,<font color="Olive">b<sub>1</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span> ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="substut1.html#NM1">CQC_Substitution</a>;<br/>

<b>set </b><font color="Maroon">c<sub>4</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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span>;<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"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></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>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span>;<br/>
<a NAME="E2:28"/><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>,<font color="Maroon">c<sub>3</sub></font></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>,<font color="Maroon">c<sub>3</sub></font></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>,<font color="Maroon">c<sub>3</sub></font></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="E3:28"/><i><font color="Green">E20</font></i>: 
( <span class="p1"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>3</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>,<font color="Maroon">c<sub>3</sub></font></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> <font color="Maroon">c<sub>3</sub></font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E4:28"/><b>then </b><i><font color="Green">E21</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>,<font color="Maroon">c<sub>3</sub></font></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>,<font color="Maroon">c<sub>3</sub></font></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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span>
 
<b>by </b><i><a class="txt" href="substut2.html#E2:28"><i><font color="Green">E19</font></i></a>, <a class="ref" href="sublemma.html#D4">SUBLEMMA:def 4</a></i>;<br/>
<a NAME="E5:28"/>
 <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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span> <a href="hidden.html#R1">=</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>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
 
<b>by </b><i><a class="txt" href="substut2.html#E3:28"><i><font color="Green">E20</font></i></a>, <a class="txt" href="substut2.html#E4:28"><i><font color="Green">E21</font></i></a>, <a class="ref" href="substut1.html#T31">SUBSTUT1:31</a></i>;<br/>
<b>then </b> <a href="cqc_sim1.html#K6">QuantNbr</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"><span class="p3">(<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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span> = 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p2">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></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"><a href="substut1.html#K39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <span class="p2">(<span class="default"><a href="substut1.html#K39">CQC_Sub</a> <span class="p3"><a href="substut2.html#K2" target="_self">[</a><span class="default"><font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E1:28"><i><font color="Green">E18</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="cqc_sim1.html#K6">QuantNbr</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
<b>by </b><i><a class="txt" href="substut2.html#E1:28"><i><font color="Green">E18</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E7:28"/>
 <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"><a href="substut1.html#K39">CQC_Sub</a> <span class="p2"><a href="substut2.html#K2" target="_self">[</a><span class="default"><span class="p3">(<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>,<font color="Maroon">c<sub>3</sub></font></span><a href="substut2.html#K2" target="_self">]</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="cqc_sim1.html#T17">CQC_SIM1:17</a></i>;<br/>


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