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<div class="add">

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<a NAME="E1:36_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="qc_lang1.html#NM3">bound_QC-variable</a><br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>5</sub></font> being  <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Olive">b<sub>4</sub></font><br/>for <font color="Olive">b<sub>6</sub></font> being  <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Olive">b<sub>4</sub></font> holds <br/>( <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K9" target="_self">VERUM</a> ] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>6</sub></font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Olive">b<sub>5</sub></font>] &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Olive">b<sub>1</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>2</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Olive">b<sub>2</sub></font>] ) &amp; ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15" target="_self">All</a> <font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>] ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a> ;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="qc_lang1.html#NM3">bound_QC-variable</a>;<br/><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="cqc_lang.html#NM2">CQC-variable_list</a> of <font color="Maroon">c<sub>4</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="qc_lang1.html#NM6">QC-pred_symbol</a> of <font color="Maroon">c<sub>4</sub></font>;<br/>











<b>thus </b><a NAME="E1:36_1_1"/>
<font color="Maroon">F<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> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <a href="cqc_lang.html#K9" target="_self">VERUM</a> 
 <b>by </b><i><a class="txt" href="cqc_lang.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cqc_lang.html#E2:36"><i><font color="Green">E21</font></i></a></i>;<br/>

<a NAME="E2:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>) &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>5</sub></font>(<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>5</sub></font>) )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cqc_lang.html#E2:36"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E3:36_1_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="cqc_lang.html#K8" target="_self">!</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/>

<a NAME="E4:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>6</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cqc_lang.html#E2:36"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E5:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> implies <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K10" target="_self">'not'</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> )
 ;<br/>

<a NAME="E6:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>) &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>7</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>) )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cqc_lang.html#E2:36"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>2</sub></font> implies <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="cqc_lang.html#K11" target="_self">'&amp;'</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> )
 ;<br/>

<a NAME="E8:36_1_1"/>
( <font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) &amp; <font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>8</sub></font>(<font color="Maroon">c<sub>3</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">F<sub>3</sub></font>() <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>) )
 
<b>by </b><i><a class="txt" href="cqc_lang.html#E1:36"><i><font color="Green">E20</font></i></a>, <a class="txt" href="cqc_lang.html#E2:36"><i><font color="Green">E21</font></i></a></i>;<br/>
<b>hence </b><a NAME="E9:36_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>] implies <font color="Maroon">S<sub>1</sub></font>[ <a href="cqc_lang.html#K15" target="_self">All</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>1</sub></font>] )
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:36_1"/>
for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="cqc_lang.html#K7" target="_self">CQC-WFF</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 
<b>from </b><i><a class="ref" href="cqc_lang.html#S1" target="_self">CQC_LANG:sch 1</a>(<a class="txt" href="cqc_lang.html#E1:36_1"><i><font color="Green">E22</font></i></a>);<br/></i>
<b>hence </b><a NAME="E3:36_1"/>
<font color="Maroon">F<sub>2</sub></font>() <a href="funct_2.html#R2">=</a> <font color="Maroon">F<sub>3</sub></font>()
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>


</div>
