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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <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> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  = <span class="p1">(<span class="default"><a href="qc_lang1.html#K15">the_pred_symbol_of</a> <font color="Maroon">a<sub>1</sub></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="Maroon">a<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  =  <a href="qc_lang1.html#K12">'not'</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  = <font color="Maroon">a<sub>1</sub></font> <a href="qc_lang1.html#K13">'&amp;'</a> <font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</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="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">a<sub>1</sub></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="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>assume </b><a NAME="E1:43"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>2</sub></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">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></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">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/>

<a NAME="E2:43"/><b>then </b><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></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="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) ) )
 
;<br/>
<b>assume </b><a NAME="E3:43"/>
for <font color="Olive">b<sub>1</sub></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">c<sub>3</sub></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">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></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">b<sub>1</sub></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">c<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="qc_lang1.html#K12">'not'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> <a href="qc_lang1.html#K13">'&amp;'</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>,<font color="Olive">b<sub>1</sub></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">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></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">b<sub>1</sub></font></span>)</span></span>)</span></span>)</span> ) )
 ;<br/>

<a NAME="E4:43"/><b>then </b><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of  <a href="qc_lang1.html#K8">QC-WFF</a>  holds <br/>( ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="cqc_lang.html#K9" target="_self">VERUM</a>  ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V2">atomic</a> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V3">negative</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K17">the_argument_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V4">conjunctive</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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="Olive">b<sub>1</sub></font></span>)</span> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></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="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>) ) &amp; ( <font color="Olive">b<sub>1</sub></font> is <a href="qc_lang1.html#V5">universal</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="qc_lang1.html#K21">the_scope_of</a> <font color="Olive">b<sub>1</sub></font></span>)</span> implies <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>) ) )
 
;<br/>
<b>thus </b><a NAME="E5:43"/>
<font color="Maroon">c<sub>2</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>from </b><i><a class="ref" href="qc_lang3.html#S1">QC_LANG3:sch 1</a>(<a class="txt" href="cqc_lang.html#E2:43"><i><font color="Green">E21</font></i></a>, <a class="txt" href="cqc_lang.html#E4:43"><i><font color="Green">E22</font></i></a>);</i><br/>


</div>
