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<b>let </b><font color="Maroon">S</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:69"/>
<font color="Maroon">S</font> is <a href="substut1.html#V6" target="_self">Sub_conjunctive</a>
 ;<br/>

<b>then consider </b><font color="Maroon">S1</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> , <font color="Maroon">S2</font> being    <a href="subset_1.html#M1">Element</a> of  <a href="substut1.html#K16" target="_self">QC-Sub-WFF</a> <b> such that </b><br/><a NAME="E3:69"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a>  <a href="substut1.html#K21" target="_self">Sub_&amp;</a> <font color="Maroon">S1</font>,<font color="Maroon">S2</font> &amp; <font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K19" target="_self">`2</a>  )
 <b>by </b><i/>;<br/>
<a NAME="E4:69"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p3">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K19" target="_self">`2</a> </span>)</span></span><a href="domain_1.html#K1">]</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E5:69"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E6:69"/><i><font color="Green">E46</font></i>: 
<span class="p1">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> 
 
<b>by </b><i>, </i>;<br/>
<a NAME="E7:69"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a>  <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span>
 
<b>by </b><i>, <a class="ref" href="mcart_1.html#T7">MCART_1:7</a></i>;<br/>
<a NAME="E8:69"/>
<span class="p1">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> is <a href="qc_lang1.html#V4">conjunctive</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D19">QC_LANG1:def 19</a></i>;<br/>
<a NAME="E9:69"/><b>then </b>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">S1</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p4">(<span class="default"><font color="Maroon">S2</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> )
 
<b>by </b><i><a class="txt" href="substut1.html#E5:69"><i><font color="Green">E45</font></i></a>, <a class="ref" href="qc_lang1.html#T46">QC_LANG1:46</a></i>;<br/>
<b>hence </b><a NAME="E10:69"/>
(  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K28" target="_self">Sub_the_left_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> &amp;  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="substut1.html#K29" target="_self">Sub_the_right_argument_of</a> <font color="Maroon">S</font></span>)</span> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K10">@</a> <span class="p2">(<span class="default"><font color="Maroon">S</font> <a href="substut1.html#K18" target="_self">`1</a> </span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="substut1.html#E4:69"><i><font color="Green">E43</font></i></a>, , <a class="ref" href="qc_lang2.html#T5">QC_LANG2:5</a></i>;<br/>


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