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<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> ] <b>means </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:1_1"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">P</font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">k</font><br/> for <font color="Olive">l</font> being   <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Olive">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Olive">l</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">k</font>;<br/><b>let </b><font color="Maroon">l</font> be    <a href="qc_lang1.html#M1">QC-variable_list</a> of <font color="Maroon">k</font>;<br/>





<a NAME="E1:1_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font> is <a href="qc_lang1.html#V2">atomic</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D17">QC_LANG1:def 17</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font></span>)</span> = 
<font color="Maroon">F<sub>5</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font></span>)</span>)
<b>by </b><i/>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="qc_lang1.html#K9">!</a> <font color="Maroon">l</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E2:1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <a href="qc_lang1.html#K11">VERUM</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>() )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E3:1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K11">VERUM</a> ]
 
;<br/>
<a NAME="E4:1_1"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12">'not'</a> <font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>

<b>assume </b><a NAME="E1:1_1_2"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font> is <a href="qc_lang1.html#V3">negative</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D18">QC_LANG1:def 18</a></i>;<br/>
<a NAME="E3:1_1_2"/><b>then </b><i><font color="Green">E50</font></i>: 
 <a href="qc_lang1.html#K17">the_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D23">QC_LANG1:def 23</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>)
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E5:1_1"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">p</font>, <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Olive">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/><b>let </b><font color="Maroon">q</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:1_1_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font> )
 ;<br/>

<a NAME="E2:1_1_3"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font> 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="E3:1_1_3"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="qc_lang1.html#K18">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="qc_lang1.html#K19">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D24">QC_LANG1:def 24</a>, <a class="ref" href="qc_lang1.html#D25">QC_LANG1:def 25</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span> = 
<font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">q</font></span>)</span>)
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">q</font></span>)</span>
<b>by </b><i>, , </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E6:1_1"/><i><font color="Green">E55</font></i>: 
for <font color="Olive">x</font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/> for <font color="Olive">p</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14">All</a> <font color="Olive">x</font>,<font color="Olive">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>assume </b><a NAME="E1:1_1_4"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E2:1_1_4"/><i><font color="Green">E57</font></i>: 
 <a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font> is <a href="qc_lang1.html#V5">universal</a>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D20">QC_LANG1:def 20</a></i>;<br/>
<a NAME="E3:1_1_4"/><b>then </b>
(  <a href="qc_lang1.html#K21">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp;  <a href="qc_lang1.html#K20">bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D26">QC_LANG1:def 26</a>, <a class="ref" href="qc_lang1.html#D27">QC_LANG1:def 27</a></i>;<br/>
<b>hence </b><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span> = 
<font color="Maroon">F<sub>8</sub></font>(<span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span></span>)</span></span>)</span>)
<b>by </b><i>, , </i>
<br/>.= 
<font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <font color="Maroon">x</font>,<font color="Maroon">p</font></span>)</span>
<b>by </b><i>, </i>
;<br/><br/>


</div><b>end;</b></div>
<a NAME="E7:1_1"/>
for <font color="Olive">p</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">p</font>]
 
<b>from </b><i><a class="ref" href="qc_lang1.html#S1">QC_LANG1:sch 1</a>(, , , , );<br/></i>
<b>hence </b><a NAME="E8:1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#R2">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
