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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b> non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  = <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>);<br/>
<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<a NAME="E1:20_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  holds   <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) is   <a href="relat_1.html#NM1">Relation</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:20_1"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/> not <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x</font></span><a href="tarski.html#K1">}</a></span> is <a href="facirc_1.html#V2">with_pair</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:20_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1) <a href="hidden.html#R1">=</a> <font color="Olive">S</font> <a href="circcomb.html#K3">+*</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">F<sub>4</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>3</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>) )
 
<b>by </b><i/>;<br/>
<a NAME="E4:20_1"/><i><font color="Green">E77</font></i>: 
( <font color="Maroon">H<sub>1</sub></font>(0) <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(0) &amp; <font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(0) )
 
<b>by </b><i/>;<br/>
<a NAME="E5:20_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S1</font>, <font color="Olive">S2</font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>) &amp; <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span>,<font color="Olive">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">S1</font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S1</font> is <a href="facirc_1.html#V2">with_pair</a> )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S10" target="_self">CIRCCMB2:sch 10</a>(, , , , , );<br/></i>
<a NAME="E6:20_1"/><b>then </b>
 ex <font color="Olive">S1</font>, <font color="Olive">S2</font> being non <a href="struct_0.html#V3">empty</a> non <a href="msualg_1.html#V2">void</a> <a href="circcomb.html#V1">unsplit</a> <a href="circcomb.html#V2">gate`1=arity</a> <a href="circcomb.html#V3">gate`2isBoolean</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  st <br/>( <font color="Olive">S1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) &amp; <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) &amp;  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S1</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Olive">S1</font> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Olive">S1</font> is <a href="facirc_1.html#V2">with_pair</a> )
 
;<br/>
<b>hence </b><a NAME="E7:20_1"/>
(  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>) <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>)</span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>3</sub></font>(<span class="p1">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span>,<font color="Maroon">n</font>)</span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">F<sub>2</sub></font>() <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="tarski.html#K1">}</a></span></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">F<sub>1</sub></font>(<font color="Maroon">n</font>) is <a href="facirc_1.html#V2">with_pair</a> )
 ;<br/>


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