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<div><i><font color="Green" title="E34">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">k</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">P</font> being  <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font><br/> for <font color="Olive" title="b2">ll</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b2">ll</font>]</span><br/><b>let </b><font color="Maroon" title="c2">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">ll</font> being   <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font> holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b1">ll</font>]</span><br/><b>let </b><font color="Maroon" title="c3">ll</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c1">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">ll</font>]</span><br/><a NAME="E1:41_1_1"/>
<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">ll</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D17" target="_self" title="QC_LANG1:def.17">Def17</a></i>;<br/><b>hence </b><a NAME="E2:41_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c3">ll</font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E2:41_1"/><i><font color="Green" title="E35">A3</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a> ]
 
<b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a></i>;<br/>
<div><i><font color="Green" title="E36">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_2"><b>now </b></a><div class="add"><b>let </b><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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font>] )</span><br/><b>assume </b><a NAME="E1:41_1_2"/><i><font color="Green" title="E37">A5</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font>]</span><br/><a NAME="E2:41_1_2"/><i><font color="Green" title="E38">A6</font></i>: 
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D18" target="_self" title="QC_LANG1:def.18">Def18</a></i>;<br/><a NAME="E3:41_1_2"/><b>then </b>
<font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D23" target="_self" title="QC_LANG1:def.23">Def23</a></i>;<br/><b>hence </b><a NAME="E4:41_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c1">p</font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_2"><i><font color="Green" title="E37">A5</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_2"><i><font color="Green" title="E38">A6</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E37">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">q</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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>] implies <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font>] )</span><br/><b>assume </b><a NAME="E1:41_1_3"/><i><font color="Green" title="E38">A8</font></i>: 
( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">q</font>] )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font>]</span><br/><a NAME="E2:41_1_3"/><i><font color="Green" title="E39">A9</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D19" target="_self" title="QC_LANG1:def.19">Def19</a></i>;<br/><a NAME="E3:41_1_3"/><b>then </b>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> &amp; <font color="Maroon" title="c2">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> )
 <b>by </b><i><a class="ref" href="qc_lang1.html#D24" target="_self" title="QC_LANG1:def.24">Def24</a>, <a class="ref" href="qc_lang1.html#D25" target="_self" title="QC_LANG1:def.25">Def25</a></i>;<br/><b>hence </b><a NAME="E4:41_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c2">q</font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_3"><i><font color="Green" title="E38">A8</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_3"><i><font color="Green" title="E39">A9</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<div><i><font color="Green" title="E38">A10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being   <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>   st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c2">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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] implies <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>] )</span><br/><b>assume </b><a NAME="E1:41_1_4"/><i><font color="Green" title="E39">A11</font></i>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>]</span><br/><a NAME="E2:41_1_4"/><i><font color="Green" title="E40">A12</font></i>: 
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D20" target="_self" title="QC_LANG1:def.20">Def20</a></i>;<br/><a NAME="E3:41_1_4"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="qc_lang1.html#D27" target="_self" title="QC_LANG1:def.27">Def27</a></i>;<br/><b>hence </b><a NAME="E4:41_1_4"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>]
 <b>by </b><i><a class="txt" href="qc_lang1.html#E1:41"><i><font color="Green" title="E33">A1</font></i></a>, <a class="txt" href="qc_lang1.html#E1:41_1_4"><i><font color="Green" title="E39">A11</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1_4"><i><font color="Green" title="E40">A12</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>thus </b><a NAME="E6:41_1"/>
for <font color="Olive" title="b1">p</font> being   <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>  holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 <b>from </b><i><a class="ref" href="qc_lang1.html#S1" target="_self" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="qc_lang1.html#E1:41_1"><i><font color="Green" title="E34">A2</font></i></a>, <a class="txt" href="qc_lang1.html#E2:41_1"><i><font color="Green" title="E35">A3</font></i></a>, <a class="txt" href="qc_lang1.html#E3:41_1"><i><font color="Green" title="E36">A4</font></i></a>, <a class="txt" href="qc_lang1.html#E4:41_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="txt" href="qc_lang1.html#E5:41_1"><i><font color="Green" title="E38">A10</font></i></a>);</i> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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