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<div><span class="lab"><font color="Green" title="E35">A2</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="53_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>)]</span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c1">x</font> be   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  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#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Olive" title="b1">p</font>)]</span><br/><span class="kw">let </span><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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)] )</span><br/><span class="kw">assume </span><a NAME="E1:53_1_1"/><span class="lab"><font color="Green" title="E36">A3</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)]</span><br/><a NAME="E2:53_1_1"/><span class="lab"><font color="Green" title="E37">A4</font></span>: 
 <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">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> 
 ;<br/><a NAME="E3:53_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D28" target="_self" onmouseover="rs('qc_lang1/D28')" onmouseout="rh()">Def28</a></span>;<br/><span class="kw">hence </span><a NAME="E4:53_1_1"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">p</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:53_1_1"><span class="lab"><font color="Green" title="E36">A3</font></span></a>, <a class="txt" href="#E2:53_1_1"><span class="lab"><font color="Green" title="E37">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E36">A5</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="53_1_2"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Olive" title="b1">p</font>]</span></a><div class="add"><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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c1">p</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:53_1_2"/><span class="lab"><font color="Green" title="E37">A6</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font>]
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c1">p</font>]</span><br/><a NAME="E2:53_1_2"/><span class="lab"><font color="Green" title="E38">A7</font></span>: 
 <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 ;<br/><a NAME="E3:53_1_2"/><span class="kw">then </span>
<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_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c1">p</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D24" target="_self" onmouseover="rs('qc_lang1/D24')" onmouseout="rh()">Def24</a></span>;<br/><span class="kw">hence </span><a NAME="E4:53_1_2"/>
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c1">p</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:53_1_2"><span class="lab"><font color="Green" title="E37">A6</font></span></a>, <a class="txt" href="#E2:53_1_2"><span class="lab"><font color="Green" title="E38">A7</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<div><span class="lab"><font color="Green" title="E37">A8</font></span>: <a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="53_1_3"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>()  st <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Olive" title="b2">q</font>]</span></a><div class="add"><span class="kw">let </span><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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>(); <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c2">q</font>] )</span><br/><span class="kw">assume </span><a NAME="E1:53_1_3"/><span class="lab"><font color="Green" title="E38">A9</font></span>: 
( <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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c2">q</font>]</span><br/><a NAME="E2:53_1_3"/><span class="lab"><font color="Green" title="E39">A10</font></span>: 
<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c2">q</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 ;<br/><a NAME="E3:53_1_3"/><span class="kw">then </span>
( <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&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#K20" title="QC_LANG1:func.20">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c2">q</font></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D25" target="_self" onmouseover="rs('qc_lang1/D25')" onmouseout="rh()">Def25</a>, <a class="ref" href="qc_lang1.html#D26" target="_self" onmouseover="rs('qc_lang1/D26')" onmouseout="rh()">Def26</a></span>;<br/><span class="kw">hence </span><a NAME="E4:53_1_3"/>
<font color="Maroon">P<sub>1</sub></font>[<font color="Maroon" title="c1">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c2">q</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="txt" href="#E1:53_1_3"><span class="lab"><font color="Green" title="E38">A9</font></span></a>, <a class="txt" href="#E2:53_1_3"><span class="lab"><font color="Green" title="E39">A10</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:53_1"/><span class="lab"><font color="Green" title="E38">A11</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon">F<sub>1</sub></font>()<br/>  for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b3">ll</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a>, <a class="ref" href="qc_lang1.html#D18" target="_self" onmouseover="rs('qc_lang1/D18')" onmouseout="rh()">Def18</a></span>;<br/>
<a NAME="E5:53_1"/><span class="lab"><font color="Green" title="E39">A12</font></span>: 
<font color="Maroon">P<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon">F<sub>1</sub></font>()]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:53"><span class="lab"><font color="Green" title="E34">A1</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:53_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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon">F<sub>1</sub></font>() holds  <font color="Maroon">P<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S1" target="_self" onmouseover="rs('qc_lang1/S1')" onmouseout="rh()">QC_LANG1:sch 1</a>(<a class="txt" href="#E4:53_1"><span class="lab"><font color="Green" title="E38">A11</font></span></a>, <a class="txt" href="#E5:53_1"><span class="lab"><font color="Green" title="E39">A12</font></span></a>, <a class="txt" href="#E2:53_1"><span class="lab"><font color="Green" title="E36">A5</font></span></a>, <a class="txt" href="#E3:53_1"><span class="lab"><font color="Green" title="E37">A8</font></span></a>, <a class="txt" href="#E1:53_1"><span class="lab"><font color="Green" title="E35">A2</font></span></a>);</span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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