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<span class="kw">let </span><font color="Maroon" title="c1">p</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> (  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c1">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><br/>

<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(   <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> ) <span class="kw">-&gt; </span>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  =  <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> $1</span>)</span>;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>2</sub></font>(   <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> ) <span class="kw">-&gt; </span>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  =  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> $1;<br/>
<span class="kw">set </span><font color="Maroon" title="c2">V</font> =  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>3</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ) <span class="kw">-&gt; </span>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  = $1;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>4</sub></font>(  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ,  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  = $1 <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> $2;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>5</sub></font>(   <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 href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> ) <span class="kw">-&gt; </span>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  = $2 <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> $1</span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E1:15"/><span class="lab"><font color="Green" title="E5">A1</font></span>: 
for <font color="Olive" title="b1">p</font> being  <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-formula</a><br/> for <font color="Olive" title="b2">X</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  holds <br/> ( <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">p</font>) iff  ex <font color="Olive" title="b3">F</font> being  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <a href="qc_lang1.html#K8" title="QC_LANG1:func.8">QC-WFF</a> ,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> </span>)</span> st <br/>( <font color="Olive" title="b2">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> &amp; ( for <font color="Olive" title="b4">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> <br/> for <font color="Olive" title="b5">d1</font>, <font color="Olive" title="b6">d2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  holds <br/> ( ( <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  implies <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) &amp; ( <font color="Olive" title="b4">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b4">p</font>) ) &amp; ( <font color="Olive" title="b4">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  &amp; <font color="Olive" title="b5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Olive" title="b4">p</font></span>)</span> implies <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b5">d1</font>) ) &amp; ( <font color="Olive" title="b4">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  &amp; <font color="Olive" title="b5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Olive" title="b4">p</font></span>)</span> &amp; <font color="Olive" title="b6">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Olive" title="b4">p</font></span>)</span> implies <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b5">d1</font>,<font color="Olive" title="b6">d2</font>) ) &amp; ( <font color="Olive" title="b4">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  &amp; <font color="Olive" title="b5">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b4">p</font></span>)</span> implies <font color="Olive" title="b3">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b4">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Olive" title="b4">p</font>,<font color="Olive" title="b5">d1</font>) ) ) ) ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="substut1.html#D8" target="_self" title="SUBSTUT1:def.8">Def8</a></span>;<br/>
<font color="Maroon">H<sub>2</sub></font>( <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a> ) = 
 <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang3.html#S3" title="QC_LANG3:sch.3">QC_LANG3:sch 3</a>(<a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E5">A1</font></span></a>)</span>
<br/>.= 
 <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 

;<br/>
<span class="kw">hence </span><a NAME="E3:15"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <a href="cqc_lang.html#K6" title="CQC_LANG:func.6">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="subset_1.html#K1" title="SUBSET_1:func.1">{}</a> <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c1">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><br/>

<span class="kw">thus </span><a NAME="E4:15"/>
( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c1">p</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_2"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K8" title="SUBSTUT1:func.8">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E2:15_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">p</font>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang3.html#S4" title="QC_LANG3:sch.4">QC_LANG3:sch 4</a>(<a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:15_2"><span class="lab"><font color="Green" title="E6">A2</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E5:15"/>
( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> ) )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_3"/><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <font color="Maroon" title="c1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span><br/>

<span class="kw">thus </span><a NAME="E2:15_3"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Maroon">H<sub>2</sub></font>( <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_argument_of</a> <font color="Maroon" title="c1">p</font>))
 <span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang3.html#S5" title="QC_LANG3:sch.5">QC_LANG3:sch 5</a>(<a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:15_3"><span class="lab"><font color="Green" title="E6">A3</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E6:15"/>
( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_4"/><span class="lab"><font color="Green" title="E6">A4</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span></span><br/>

<a NAME="E2:15_4"/>
for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>   st <font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</font>) &amp; <font color="Olive" title="b2">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>2</sub></font>( <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font>) holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive" title="b1">d1</font>,<font color="Olive" title="b2">d2</font>)
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang3.html#S6" title="QC_LANG3:sch.6">QC_LANG3:sch 6</a>(<a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:15_4"><span class="lab"><font color="Green" title="E6">A4</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E3:15_4"/>
 <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_left_argument_of</a> <font color="Maroon" title="c1">p</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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_right_argument_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">thus </span><a NAME="E7:15"/>
( <font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a>  implies  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:15_5"/><span class="lab"><font color="Green" title="E6">A5</font></span>: 
<font color="Maroon" title="c1">p</font> is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <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="substut1.html#K9" title="SUBSTUT1:func.9">Bound_Vars</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Maroon" title="c1">p</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span><br/>

<span class="kw">thus </span><a NAME="E2:15_5"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c1">p</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>5</sub></font>(<font color="Maroon" title="c1">p</font>,<font color="Maroon">H<sub>2</sub></font>( <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Maroon" title="c1">p</font>))
 <span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang3.html#S7" title="QC_LANG3:sch.7">QC_LANG3:sch 7</a>(<a class="txt" href="#E1:15"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:15_5"><span class="lab"><font color="Green" title="E6">A5</font></span></a>);</span> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>


</div>
