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<b>let </b><font color="Maroon" title="c2">IT</font>, <font color="Maroon" title="c3">IT'</font> be   <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 class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  ex <font color="Olive" title="b1">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> , <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>  st <br/>( <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> &amp; ( 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>  holds <br/> ( <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="b2">p</font></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b2">p</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"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) ) ) ) &amp;  ex <font color="Olive" title="b1">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> , <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>  st <br/>( <font color="Maroon" title="c3">IT'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> &amp; ( 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>  holds <br/> ( <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="b2">p</font></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b2">p</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"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) ) ) ) implies <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT'</font> )</span><br/>

<b>given </b><font color="Maroon" title="c4">F1</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> , <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> <b> such that </b><a NAME="E2:48_1_2"/><i><font color="Green" title="E37">A2</font></i>: 
<font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E3:48_1_2"/><i><font color="Green" title="E38">A3</font></i>: 
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 <br/> ( <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> implies <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">F1</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="b1">p</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> implies <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> implies <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b1">p</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"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ; <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">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> , <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>   holds <br/>(  not <font color="Maroon" title="c3">IT'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font> or  ex <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 <br/>( <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font></span>)</span> where <font color="Olive" title="b3">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">k</font> &amp; <font color="Olive" title="b3">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b2">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b3">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">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="b2">p</font></span>)</span> ) &amp; ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> implies <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b2">p</font></span>)</span></span>)</span> ) implies ( <font color="Olive" title="b2">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> &amp; not <font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">F</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b2">p</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"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Olive" title="b2">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) ) ) or <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT'</font> )</span><br/>

<b>given </b><font color="Maroon" title="c5">F2</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> , <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> <b> such that </b><a NAME="E5:48_1_2"/><i><font color="Green" title="E39">A4</font></i>: 
<font color="Maroon" title="c3">IT'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c1">p</font>
 <b>and </b><br/><a NAME="E6:48_1_2"/><i><font color="Green" title="E40">A5</font></i>: 
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 <br/> ( <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="qc_lang1.html#K11" title="QC_LANG1:func.11">VERUM</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> implies <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font></span>)</span> where <font color="Olive" title="b2">k</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">k</font> &amp; <font color="Olive" title="b2">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <font color="Olive" title="b1">p</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span>  ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> implies <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</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="b1">p</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> implies <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</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="Olive" title="b1">p</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive" title="b1">p</font> is <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> implies <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K21" title="QC_LANG1:func.21">the_scope_of</a> <font color="Olive" title="b1">p</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"><span class="p2">(<span class="default"><a href="qc_lang1.html#K20" title="QC_LANG1:func.20">bound_in</a> <font color="Olive" title="b1">p</font></span>)</span></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> ) )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT'</font></span><br/>

<b>defpred </b><font color="Maroon">S<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> ] <b>means </b><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E7:48_1_2"/><i><font color="Green" title="E41">A6</font></i>: 
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#M1" title="QC_LANG1:mode.1">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="48_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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="c6">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="c6">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/><b>let </b><font color="Maroon" title="c7">P</font> be   <a href="qc_lang1.html#NM6" title="QC_LANG1:NM.6">QC-pred_symbol</a> of <font color="Maroon" title="c6">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="c6">k</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="c8">ll</font> be    <a href="qc_lang1.html#M1" title="QC_LANG1:mode.1">QC-variable_list</a> of <font color="Maroon" title="c6">k</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c8">ll</font>]</span><br/>





<a NAME="E1:48_1_2_1"/><i><font color="Green" title="E42">A7</font></i>: 
<font color="Maroon" title="c7">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c8">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/>
<a NAME="E2:48_1_2_1"/><b>then </b><i><font color="Green" title="E43">A8</font></i>: 
 <a href="qc_lang1.html#K16" title="QC_LANG1:func.16">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c8">ll</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">ll</font>
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D22" target="_self" title="QC_LANG1:def.22">Def22</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c8">ll</font></span>)</span> = 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c8">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font></span>)</span> where <font color="Olive" title="b1">j</font> is    <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>  : ( 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c8">ll</font> &amp; <font color="Maroon" title="c8">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="qc_lang1.html#K2" title="QC_LANG1:func.2">bound_QC-variables</a>  ) </span>}</span> 
<b>by </b><i><a class="txt" href="#E3:48_1_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:48_1_2_1"><i><font color="Green" title="E42">A7</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">P</font> <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">!</a> <font color="Maroon" title="c8">ll</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E1:48_1_2_1"><i><font color="Green" title="E42">A7</font></i></a>, <a class="txt" href="#E2:48_1_2_1"><i><font color="Green" title="E43">A8</font></i></a></i>
;<br/> <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="E8:48_1_2"/><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon">S<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="#E3:48_1_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E40">A5</font></i></a></i>;<br/>
<a NAME="E9:48_1_2"/><i><font color="Green" title="E43">A10</font></i>: 
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="48_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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">S<sub>1</sub></font>[<font color="Maroon" title="c6">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="c6">p</font>] )</span><br/>

<b>assume </b><a NAME="E1:48_1_2_2"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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="c6">p</font>]</span><br/>

<a NAME="E2:48_1_2_2"/><i><font color="Green" title="E45">A12</font></i>: 
 <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c6">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="E3:48_1_2_2"/><b>then </b><i><font color="Green" title="E46">A13</font></i>: 
 <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="c6">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font>
 
<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><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c6">p</font></span>)</span> = 
<font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font>
<b>by </b><i><a class="txt" href="#E3:48_1_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:48_1_2_2"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="#E2:48_1_2_2"><i><font color="Green" title="E45">A12</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K12" title="QC_LANG1:func.12">'not'</a> <font color="Maroon" title="c6">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E2:48_1_2_2"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="#E3:48_1_2_2"><i><font color="Green" title="E46">A13</font></i></a></i>
;<br/> <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="E10:48_1_2"/><i><font color="Green" title="E44">A14</font></i>: 
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="48_1_2_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">p</font>, <font color="Maroon" title="c7">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">S<sub>1</sub></font>[<font color="Maroon" title="c6">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font>] )</span><br/>

<b>assume </b><a NAME="E1:48_1_2_3"/><i><font color="Green" title="E45">A15</font></i>: 
( <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font> &amp; <font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font> )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font>]</span><br/>

<a NAME="E2:48_1_2_3"/><i><font color="Green" title="E46">A16</font></i>: 
<font color="Maroon" title="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">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="E3:48_1_2_3"/><b>then </b><i><font color="Green" title="E47">A17</font></i>: 
(  <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="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">p</font> &amp;  <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="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">q</font> )
 
<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><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c6">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E3:48_1_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:48_1_2_3"><i><font color="Green" title="E45">A15</font></i></a>, <a class="txt" href="#E2:48_1_2_3"><i><font color="Green" title="E46">A16</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'&amp;'</a> <font color="Maroon" title="c7">q</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E2:48_1_2_3"><i><font color="Green" title="E46">A16</font></i></a>, <a class="txt" href="#E3:48_1_2_3"><i><font color="Green" title="E47">A17</font></i></a></i>
;<br/> <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="E11:48_1_2"/><i><font color="Green" title="E45">A18</font></i>: 
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="48_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">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">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="c6">x</font>,<font color="Olive" title="b1">p</font>]</span><br/><b>let </b><font color="Maroon" title="c7">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">S<sub>1</sub></font>[<font color="Maroon" title="c7">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="c6">x</font>,<font color="Maroon" title="c7">p</font>] )</span><br/>



<b>assume </b><a NAME="E1:48_1_2_4"/><i><font color="Green" title="E46">A19</font></i>: 
<font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></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="c6">x</font>,<font color="Maroon" title="c7">p</font>]</span><br/>

<a NAME="E2:48_1_2_4"/><i><font color="Green" title="E47">A20</font></i>: 
 <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">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="E3:48_1_2_4"/><b>then </b><i><font color="Green" title="E48">A21</font></i>: 
(  <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="c6">x</font>,<font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font> &amp;  <a href="qc_lang1.html#K20" title="QC_LANG1:func.20">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="c6">x</font>,<font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">x</font> )
 
<b>by </b><i><a class="ref" href="qc_lang1.html#D26" target="_self" title="QC_LANG1:def.26">Def26</a>, <a class="ref" href="qc_lang1.html#D27" target="_self" title="QC_LANG1:def.27">Def27</a></i>;<br/>
<b>hence </b><font color="Maroon" title="c4">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">p</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">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="c6">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="#E3:48_1_2"><i><font color="Green" title="E38">A3</font></i></a>, <a class="txt" href="#E1:48_1_2_4"><i><font color="Green" title="E46">A19</font></i></a>, <a class="txt" href="#E2:48_1_2_4"><i><font color="Green" title="E47">A20</font></i></a></i>
<br/>.= 
<font color="Maroon" title="c5">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K14" title="QC_LANG1:func.14">All</a> <font color="Maroon" title="c6">x</font>,<font color="Maroon" title="c7">p</font></span>)</span>
<b>by </b><i><a class="txt" href="#E6:48_1_2"><i><font color="Green" title="E40">A5</font></i></a>, <a class="txt" href="#E2:48_1_2_4"><i><font color="Green" title="E47">A20</font></i></a>, <a class="txt" href="#E3:48_1_2_4"><i><font color="Green" title="E48">A21</font></i></a></i>
;<br/> <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="E12:48_1_2"/>
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">S<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="#E7:48_1_2"><i><font color="Green" title="E41">A6</font></i></a>, <a class="txt" href="#E8:48_1_2"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="#E9:48_1_2"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="#E10:48_1_2"><i><font color="Green" title="E44">A14</font></i></a>, <a class="txt" href="#E11:48_1_2"><i><font color="Green" title="E45">A18</font></i></a>);<br/></i>
<b>hence </b><a NAME="E13:48_1_2"/>
<font color="Maroon" title="c2">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">IT'</font>
 <b>by </b><i><a class="txt" href="#E2:48_1_2"><i><font color="Green" title="E37">A2</font></i></a>, <a class="txt" href="#E5:48_1_2"><i><font color="Green" title="E39">A4</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>
