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<b>let </b><font color="Maroon">x</font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">p</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ;<br/>



<b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E2:50"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 <b>and </b><br/><a NAME="E3:50"/><i><font color="Green">E43</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/> ( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Olive">q</font></span>)</span> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4" target="_self">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2" target="_self">IFEQ</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<font color="Maroon">x</font>,<font color="Olive">q</font>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">q</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F1</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E5:50"/><i><font color="Green">E51</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E6:50"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/> ( <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Olive">q</font></span>)</span> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4" target="_self">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2" target="_self">IFEQ</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<font color="Maroon">x</font>,<font color="Olive">q</font>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F1</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">q</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">F2</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> , <a href="qc_lang1.html#K8">QC-WFF</a> <b> such that </b><br/><a NAME="E8:50"/><i><font color="Green">E53</font></i>: 
<span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">p</font></span>)</span>
 <b>and </b><br/><a NAME="E9:50"/><i><font color="Green">E54</font></i>: 
for <font color="Olive">q</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/> ( <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a>  <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Olive">q</font></span>)</span> <a href="qc_lang1.html#K9">!</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K4" target="_self">Subst</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K16">the_arguments_of</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="qc_lang3.html#K3">a.</a> 0</span>)</span> <a href="cqc_lang.html#K6" target="_self">.--&gt;</a> <font color="Maroon">x</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Olive">q</font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">q</font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <font color="Olive">q</font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K2" target="_self">IFEQ</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<font color="Maroon">x</font>,<font color="Olive">q</font>,<span class="p1">(<span class="default"><a href="qc_lang1.html#K14">All</a> <span class="p2">(<span class="default"><a href="qc_lang1.html#K20">bound_in</a> <font color="Olive">q</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">F2</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">q</font></span>)</span></span>)</span></span>)</span> ) )
 <b>by </b><i/>;<br/>
<a NAME="E10:50"/>
( <font color="Maroon">F1</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> &amp; <font color="Maroon">F2</font> <a href="funct_2.html#R2">=</a> <font color="Maroon">F</font> )
 
<b>by </b><i><a class="ref" href="cqc_lang.html#T5" target="_self">Th5</a>, , , </i>;<br/>
<b>hence </b><a NAME="E11:50"/>
( <font color="Maroon">p</font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">p</font> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K18">the_left_argument_of</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="qc_lang1.html#K19">the_right_argument_of</a> <font color="Maroon">p</font></span>)</span> <a href="cqc_lang.html#K17" target="_self">.</a> <font color="Maroon">x</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="cqc_lang.html#T5" target="_self">Th5</a>, <a class="ref" href="cqc_lang.html#T6" target="_self">Th6</a>, </i>;<br/>


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