<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">h</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> holds   <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Olive" title="b1">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">h</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span><br/>

<span class="kw">defpred </span><font color="Maroon">S<sub>1</sub></font>[  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>] <span class="kw">means </span> <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> $1 <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default">$1 <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span>;<br/>
<a NAME="E1:69"/><span class="lab"><font color="Green" title="E54">A1</font></span>: 
 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">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1"><span class="kw">proof </span></a><div class="add">

<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#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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">p</font>] )</span><br/>

 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T19" title="CQC_LANG:th.19">CQC_LANG:19</a></span>
<br/>.= 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T7" title="QC_LANG3:th.7">QC_LANG3:7</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E2:69_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c2">p</font>] )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T7" title="QC_LANG3:th.7">QC_LANG3:7</a></span>; <a class="txt" 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="E2:69"/><span class="lab"><font color="Green" title="E55">A2</font></span>: 
 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#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">q</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">q</font>] )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:69_2"/><span class="lab"><font color="Green" title="E56">A3</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font>]
 <span class="kw">and </span><br/><a NAME="E2:69_2"/><span class="lab"><font color="Green" title="E57">A4</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">q</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">q</font>]</span><br/>

<span class="lab"><font color="Green" title="E58">A5</font></span>:  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T21" title="CQC_LANG:th.21">CQC_LANG:21</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</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_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">q</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T10" title="QC_LANG3:th.10">QC_LANG3:10</a></span>
;<br/>
<a NAME="E4:69_2"/>
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c3">q</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T10" title="QC_LANG3:th.10">QC_LANG3:10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:69_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c2">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c3">q</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_2"><span class="lab"><font color="Green" title="E56">A3</font></span></a>, <a class="txt" href="#E2:69_2"><span class="lab"><font color="Green" title="E57">A4</font></span></a>, <a class="txt" href="#E3:69_2"><span class="lab"><font color="Green" title="E58">A5</font></span></a>, <a class="ref" href="xboole_1.html#T13" title="XBOOLE_1:th.13">XBOOLE_1:13</a></span>; <a class="txt" 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="E3:69"/><span class="lab"><font color="Green" title="E56">A6</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM3" title="QC_LANG1:NM.3">bound_QC-variable</a><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#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>)]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">y</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:()"><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#K8" title="QC_LANG1:func.8">QC-WFF</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Olive" title="b1">p</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c3">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"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:69_3"/><span class="lab"><font color="Green" title="E57">A7</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">p</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)]</span><br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:69_3_1"/>
( <font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font> or <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_1"><span class="kw">suppose </span></a><a NAME="E1:69_3_1_1"/>
<font color="Maroon" title="c2">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)]</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:69_3_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)]
 <span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T24" title="CQC_LANG:th.24">CQC_LANG:24</a></span>; <a class="txt" 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><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_3_1_2"><span class="kw">suppose </span></a><a NAME="E1:69_3_1_2"/><span class="lab"><font color="Green" title="E58">A8</font></span>: 
<font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">x</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)]</span><br/><div class="add"><a NAME="E2:69_3_1_2"/><span class="lab"><font color="Green" title="E59">A9</font></span>: 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <font color="Maroon" title="c3">p</font></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T12" title="QC_LANG3:th.12">QC_LANG3:12</a></span>;<br/> <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)</span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_3_1_2"><span class="lab"><font color="Green" title="E58">A8</font></span></a>, <a class="ref" href="cqc_lang.html#T25" title="CQC_LANG:th.25">CQC_LANG:25</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span></span>)</span> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c2">y</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T12" title="QC_LANG3:th.12">QC_LANG3:12</a></span>
;<br/><span class="kw">hence </span><a NAME="E4:69_3_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> (<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">p</font>)]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_3"><span class="lab"><font color="Green" title="E57">A7</font></span></a>, <a class="txt" href="#E2:69_3_1_2"><span class="lab"><font color="Green" title="E59">A9</font></span></a>, <a class="ref" href="xboole_1.html#T33" title="XBOOLE_1:th.33">XBOOLE_1:33</a></span>; <a class="txt" 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="kw">end;</span></div>

</div><span class="kw">end;</span></div>
<a NAME="E4:69"/><span class="lab"><font color="Green" title="E57">A10</font></span>: 
 for <font color="Olive" title="b1">k</font> being    <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> <br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font><br/>  for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Olive" title="b1">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b3">ll</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c2">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c2">k</font><br/>  for <font color="Olive" title="b2">ll</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font> holds  <font color="Maroon">S<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/><span class="kw">let </span><font color="Maroon" title="c3">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c2">k</font>; <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">ll</font> being   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Olive" title="b1">ll</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c4">ll</font> be   <a href="qc_lang1.html#NM7" title="QC_LANG1:NM.7">QC-variable_list</a> of <font color="Maroon" title="c2">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">ll</font>]</span><br/>





<span class="lab"><font color="Green" title="E58">A11</font></span>:  <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">ll</font></span>)</span> <a href="cqc_lang.html#K13" title="CQC_LANG:func.13">.</a> <font color="Maroon" title="c1">x</font></span>)</span> = 
 <a href="qc_lang1.html#K23" title="QC_LANG1:func.23">still_not-bound_in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <span class="p2">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c4">ll</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T17" title="CQC_LANG:th.17">CQC_LANG:17</a></span>
<br/>.= 
 <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c4">ll</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang3.html#T5" title="QC_LANG3:th.5">QC_LANG3:5</a></span>
;<br/>
<a NAME="E2:69_4"/>
 <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <font color="Maroon" title="c4">ll</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">still_not-bound_in</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K1" title="CQC_LANG:func.1">Subst</a> (<font color="Maroon" title="c4">ll</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3" title="QC_LANG3:func.3">a.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="cqc_lang.html#K2" title="CQC_LANG:func.2">.--&gt;</a> <font color="Maroon" title="c1">x</font></span>)</span>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_the3.html#T60" target="_self" title="CQC_THE3:th.60">Th61</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:69_4"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c3">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c4">ll</font>]
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:69_4"><span class="lab"><font color="Green" title="E58">A11</font></span></a>, <a class="ref" href="qc_lang3.html#T5" title="QC_LANG3:th.5">QC_LANG3:5</a></span>; <a class="txt" 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="E5:69"/><span class="lab"><font color="Green" title="E58">A12</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K5" title="CQC_LANG:func.5">VERUM</a> ]
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cqc_lang.html#T15" title="CQC_LANG:th.15">CQC_LANG:15</a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:69"/>
 for <font color="Olive" title="b1">h</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">h</font>]
 <span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S1" title="QC_LANG1:sch.1">QC_LANG1:sch 1</a>(<a class="txt" href="#E4:69"><span class="lab"><font color="Green" title="E57">A10</font></span></a>, <a class="txt" href="#E5:69"><span class="lab"><font color="Green" title="E58">A12</font></span></a>, <a class="txt" href="#E1:69"><span class="lab"><font color="Green" title="E54">A1</font></span></a>, <a class="txt" href="#E2:69"><span class="lab"><font color="Green" title="E55">A2</font></span></a>, <a class="txt" href="#E3:69"><span class="lab"><font color="Green" title="E56">A6</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
