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<span class="kw">let </span><font color="Maroon" title="c3">IT</font>, <font color="Maroon" title="c4">IT9</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  ex <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> 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="c2">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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> holds <br/> ( <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#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font></span>)</span> <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#K17" title="QC_LANG1:func.17">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K17" title="QC_LANG1:func.17">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#K17" title="QC_LANG1:func.17">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </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#K18" title="QC_LANG1:func.18">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#K19" title="QC_LANG1:func.19">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#K20" title="QC_LANG1:func.20">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#K22" title="QC_LANG1:func.22">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#K21" title="QC_LANG1:func.21">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 <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span> st <br/>( <font color="Maroon" title="c4">IT9</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="c2">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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> holds <br/> ( <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#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font></span>)</span> <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#K17" title="QC_LANG1:func.17">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K17" title="QC_LANG1:func.17">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#K17" title="QC_LANG1:func.17">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </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#K18" title="QC_LANG1:func.18">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#K19" title="QC_LANG1:func.19">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#K20" title="QC_LANG1:func.20">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#K22" title="QC_LANG1:func.22">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#K21" title="QC_LANG1:func.21">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="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT9</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c5">F1</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E2:60_1_2"/><span class="lab"><font color="Green" title="E38">A2</font></span>: 
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E3:60_1_2"/><span class="lab"><font color="Green" title="E39">A3</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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon" title="c5">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">VERUM</a> <font color="Maroon" title="c1">A</font></span>)</span> <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">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#K17" title="QC_LANG1:func.17">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K17" title="QC_LANG1:func.17">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#K17" title="QC_LANG1:func.17">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </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">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="c5">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#K18" title="QC_LANG1:func.18">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">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="c5">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_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">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#K20" title="QC_LANG1:func.20">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">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="c5">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#K22" title="QC_LANG1:func.22">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#K21" title="QC_LANG1:func.21">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  for <font color="Olive" title="b1">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span>  holds <br/>(  not <font color="Maroon" title="c4">IT9</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="c2">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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> st <br/>( <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#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font></span>)</span> <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#K17" title="QC_LANG1:func.17">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K17" title="QC_LANG1:func.17">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#K17" title="QC_LANG1:func.17">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </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#K18" title="QC_LANG1:func.18">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#K19" title="QC_LANG1:func.19">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#K20" title="QC_LANG1:func.20">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#K22" title="QC_LANG1:func.22">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#K21" title="QC_LANG1:func.21">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="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT9</font> )</span><br/>

<span class="kw">given </span><font color="Maroon" title="c6">F2</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <span class="p1">(<span class="default"><a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font></span>)</span>,<span class="p1">(<span class="default"><a href="zfmisc_1.html#K1" title="ZFMISC_1:func.1">bool</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font></span>)</span></span>)</span><span class="kw"> such that </span><a NAME="E5:60_1_2"/><span class="lab"><font color="Green" title="E40">A4</font></span>: 
<font color="Maroon" title="c4">IT9</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">p</font>
 <span class="kw">and </span><br/><a NAME="E6:60_1_2"/><span class="lab"><font color="Green" title="E41">A5</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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> holds <br/> ( <font color="Maroon" title="c6">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">VERUM</a> <font color="Maroon" title="c1">A</font></span>)</span> <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="c6">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#K17" title="QC_LANG1:func.17">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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#K17" title="QC_LANG1:func.17">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#K17" title="QC_LANG1:func.17">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="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </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="c6">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="c6">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#K18" title="QC_LANG1:func.18">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="c6">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="c6">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_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="c6">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#K20" title="QC_LANG1:func.20">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="c6">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="c6">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#K22" title="QC_LANG1:func.22">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#K21" title="QC_LANG1:func.21">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT9</font></span><br/>

<span class="kw">defpred </span><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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>] <span class="kw">means </span><font color="Maroon" title="c5">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="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1;<br/>
<a NAME="E7:60_1_2"/><span class="lab"><font color="Green" title="E42">A6</font></span>: 
 for <font color="Olive" title="b1">k</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a><br/>  for <font color="Olive" title="b2">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b3">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Olive" title="b1">k</font>,<font color="Maroon" title="c1">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b2">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b3">ll</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">k</font> be   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">P</font> being   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b2">ll</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c8">P</font> be   <a href="qc_lang1.html#NM8" title="QC_LANG1:NM.8">QC-pred_symbol</a> of <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">ll</font> being   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Olive" title="b1">ll</font>]</span><br/><span class="kw">let </span><font color="Maroon" title="c9">ll</font> be   <a href="qc_lang1.html#NM9" title="QC_LANG1:NM.9">QC-variable_list</a> of <font color="Maroon" title="c7">k</font>,<font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c9">ll</font>]</span><br/>





<a NAME="E1:60_1_2_1"/><span class="lab"><font color="Green" title="E43">A7</font></span>: 
<font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c9">ll</font> is  <a href="qc_lang1.html#V2" title="QC_LANG1:attr.2">atomic</a> 
 
;<br/>
<a NAME="E2:60_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E44">A8</font></span>: 
 <a href="qc_lang1.html#K17" title="QC_LANG1:func.17">the_arguments_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c9">ll</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">ll</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D23" target="_self" onmouseover="rs('qc_lang1/D23')" onmouseout="rh()">Def23</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c5">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c9">ll</font></span>)</span> = 
<span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c9">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="ordinal1.html#NM6" title="ORDINAL1:NM.6">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="c9">ll</font> &amp; <font color="Maroon" title="c9">ll</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">j</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="qc_lang1.html#K3" title="QC_LANG1:func.3">bound_QC-variables</a> <font color="Maroon" title="c1">A</font> ) </span> } </span> 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:60_1_2_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">P</font> <a href="qc_lang1.html#K10" title="QC_LANG1:func.10">!</a> <font color="Maroon" title="c9">ll</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E1:60_1_2_1"><span class="lab"><font color="Green" title="E43">A7</font></span></a>, <a class="txt" href="#E2:60_1_2_1"><span class="lab"><font color="Green" title="E44">A8</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:60_1_2"/><span class="lab"><font color="Green" title="E43">A9</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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>  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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Olive" title="b1">p</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_2_2"/><span class="lab"><font color="Green" title="E44">A10</font></span>: 
<font color="Maroon" title="c5">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="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font>]</span><br/>

<a NAME="E2:60_1_2_2"/><span class="lab"><font color="Green" title="E45">A11</font></span>: 
 <a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font> is  <a href="qc_lang1.html#V3" title="QC_LANG1:attr.3">negative</a> 
 
;<br/>
<a NAME="E3:60_1_2_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E46">A12</font></span>: 
 <a href="qc_lang1.html#K18" title="QC_LANG1:func.18">the_argument_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D24" target="_self" onmouseover="rs('qc_lang1/D24')" onmouseout="rh()">Def24</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c5">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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font></span>)</span> = 
<font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:60_1_2_2"><span class="lab"><font color="Green" title="E44">A10</font></span></a>, <a class="txt" href="#E2:60_1_2_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">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#K13" title="QC_LANG1:func.13">'not'</a> <font color="Maroon" title="c7">p</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E2:60_1_2_2"><span class="lab"><font color="Green" title="E45">A11</font></span></a>, <a class="txt" href="#E3:60_1_2_2"><span class="lab"><font color="Green" title="E46">A12</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E9:60_1_2"/><span class="lab"><font color="Green" title="E44">A13</font></span>: 
 for <font color="Olive" title="b1">x</font> being   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font><br/>  for <font color="Olive" title="b2">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>  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#K15" title="QC_LANG1:func.15">All</a> (<font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">p</font>)]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">x</font> be   <a href="qc_lang1.html#NM5" title="QC_LANG1:NM.5">bound_QC-variable</a> of <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>  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#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Olive" title="b1">p</font>)]</span><br/><span class="kw">let </span><font color="Maroon" title="c8">p</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">p</font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)] )</span><br/>



<span class="kw">assume </span><a NAME="E1:60_1_2_3"/><span class="lab"><font color="Green" title="E45">A14</font></span>: 
<font color="Maroon" title="c5">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">p</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)]</span><br/>

<a NAME="E2:60_1_2_3"/><span class="lab"><font color="Green" title="E46">A15</font></span>: 
 <a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>) is  <a href="qc_lang1.html#V5" title="QC_LANG1:attr.5">universal</a> 
 
;<br/>
<a NAME="E3:60_1_2_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A16</font></span>: 
(  <a href="qc_lang1.html#K22" title="QC_LANG1:func.22">the_scope_of</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">p</font> &amp;  <a href="qc_lang1.html#K21" title="QC_LANG1:func.21">bound_in</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">x</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D27" target="_self" onmouseover="rs('qc_lang1/D27')" onmouseout="rh()">Def27</a>, <a class="ref" href="qc_lang1.html#D28" target="_self" onmouseover="rs('qc_lang1/D28')" onmouseout="rh()">Def28</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c5">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#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">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="c7">x</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:60_1_2_3"><span class="lab"><font color="Green" title="E45">A14</font></span></a>, <a class="txt" href="#E2:60_1_2_3"><span class="lab"><font color="Green" title="E46">A15</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">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#K15" title="QC_LANG1:func.15">All</a> (<font color="Maroon" title="c7">x</font>,<font color="Maroon" title="c8">p</font>)</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E2:60_1_2_3"><span class="lab"><font color="Green" title="E46">A15</font></span></a>, <a class="txt" href="#E3:60_1_2_3"><span class="lab"><font color="Green" title="E47">A16</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E10:60_1_2"/><span class="lab"><font color="Green" title="E45">A17</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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>  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#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Olive" title="b2">q</font>]
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="60_1_2_4"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c7">p</font>, <font color="Maroon" title="c8">q</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="qc_lang1.html#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">p</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c8">q</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font>] )</span><br/>

<span class="kw">assume </span><a NAME="E1:60_1_2_4"/><span class="lab"><font color="Green" title="E46">A18</font></span>: 
( <font color="Maroon" title="c5">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="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">p</font> &amp; <font color="Maroon" title="c5">F1</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font>]</span><br/>

<a NAME="E2:60_1_2_4"/><span class="lab"><font color="Green" title="E47">A19</font></span>: 
<font color="Maroon" title="c7">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font> is  <a href="qc_lang1.html#V4" title="QC_LANG1:attr.4">conjunctive</a> 
 
;<br/>
<a NAME="E3:60_1_2_4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E48">A20</font></span>: 
(  <a href="qc_lang1.html#K19" title="QC_LANG1:func.19">the_left_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</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">the_right_argument_of</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">p</font> <a href="qc_lang1.html#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">q</font> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="qc_lang1.html#D25" target="_self" onmouseover="rs('qc_lang1/D25')" onmouseout="rh()">Def25</a>, <a class="ref" href="qc_lang1.html#D26" target="_self" onmouseover="rs('qc_lang1/D26')" onmouseout="rh()">Def26</a></span>;<br/>
<span class="kw">hence </span><font color="Maroon" title="c5">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#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font></span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">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#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">F2</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E1:60_1_2_4"><span class="lab"><font color="Green" title="E46">A18</font></span></a>, <a class="txt" href="#E2:60_1_2_4"><span class="lab"><font color="Green" title="E47">A19</font></span></a></span>
<br/>.= 
<font color="Maroon" title="c6">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#K14" title="QC_LANG1:func.14">'&amp;'</a> <font color="Maroon" title="c8">q</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:60_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a>, <a class="txt" href="#E2:60_1_2_4"><span class="lab"><font color="Green" title="E47">A19</font></span></a>, <a class="txt" href="#E3:60_1_2_4"><span class="lab"><font color="Green" title="E48">A20</font></span></a></span>
;<br/> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E11:60_1_2"/><span class="lab"><font color="Green" title="E46">A21</font></span>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="qc_lang1.html#K12" title="QC_LANG1:func.12">VERUM</a> <font color="Maroon" title="c1">A</font>]
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:60_1_2"><span class="lab"><font color="Green" title="E39">A3</font></span></a>, <a class="txt" href="#E6:60_1_2"><span class="lab"><font color="Green" title="E41">A5</font></span></a></span>;<br/>
<a NAME="E12:60_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#K9" title="QC_LANG1:func.9">QC-WFF</a> <font color="Maroon" title="c1">A</font> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">p</font>]
 
<span class="kw">from</span> <span class="lab"><a class="ref" href="qc_lang1.html#S1" target="_self" onmouseover="rs('qc_lang1/S1')" onmouseout="rh()">QC_LANG1:sch 1</a>(<a class="txt" href="#E7:60_1_2"><span class="lab"><font color="Green" title="E42">A6</font></span></a>, <a class="txt" href="#E11:60_1_2"><span class="lab"><font color="Green" title="E46">A21</font></span></a>, <a class="txt" href="#E8:60_1_2"><span class="lab"><font color="Green" title="E43">A9</font></span></a>, <a class="txt" href="#E10:60_1_2"><span class="lab"><font color="Green" title="E45">A17</font></span></a>, <a class="txt" href="#E9:60_1_2"><span class="lab"><font color="Green" title="E44">A13</font></span></a>);<br/></span>
<span class="kw">hence </span><a NAME="E13:60_1_2"/>
<font color="Maroon" title="c3">IT</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">IT9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:60_1_2"><span class="lab"><font color="Green" title="E38">A2</font></span></a>, <a class="txt" href="#E5:60_1_2"><span class="lab"><font color="Green" title="E40">A4</font></span></a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
