<?xml version="1.0"?>
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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">V</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-variables</a>  st <font color="Maroon" title="c1">p</font> is  <a href="qc_lang2.html#V3" title="QC_LANG2:attr.3">biconditional</a>  holds <br/> <a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">V</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Olive" title="b1">V</font>)</span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">V</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K1" title="QC_LANG1:func.1">QC-variables</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang2.html#V3" title="QC_LANG2:attr.3">biconditional</a>  implies  <a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">V</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> )</span><br/>



<span class="kw">set </span><font color="Maroon" title="c3">p1</font> =  <a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c4">p2</font> =  <a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font>;<br/>
<span class="kw">assume </span><a NAME="E1:55"/>
<font color="Maroon" title="c1">p</font> is  <a href="qc_lang2.html#V3" title="QC_LANG2:attr.3">biconditional</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">V</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span></span><br/>

<a NAME="E2:55"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K4" title="QC_LANG2:func.4">&lt;=&gt;</a> <span class="p1">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#T39" title="QC_LANG2:th.39">QC_LANG2:39</a></span>;<br/>
<a NAME="E3:55"/><span class="kw">then </span>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang2.html#D4" title="QC_LANG2:def.4">QC_LANG2:def 4</a></span>;<br/>
<span class="kw">hence </span> <a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">V</font>) = 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T42" target="_self" title="QC_LANG3:th.42">Th53</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span> <a href="qc_lang2.html#K2" title="QC_LANG2:func.2">=&gt;</a> <span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T48" target="_self" title="QC_LANG3:th.48">Th59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p2">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p3">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T48" target="_self" title="QC_LANG3:th.48">Th59</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K13" title="QC_LANG2:func.13">the_left_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang3.html#K4" title="QC_LANG3:func.4">Vars</a> (<span class="p2">(<span class="default"><a href="qc_lang2.html#K14" title="QC_LANG2:func.14">the_right_side_of</a> <font color="Maroon" title="c1">p</font></span>)</span>,<font color="Maroon" title="c2">V</font>)</span>)</span>

;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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