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<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ] <b>means </b><font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of  <a href="msualg_1.html#K4">Result</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">SCS</font> = <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>,<font color="Maroon">SCS</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s</font></span>)</span>;<br/>
<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font><b> such that </b><br/><a NAME="E2:17_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">v</font> being  <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>  st <font color="Olive">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font> holds <br/>( ( <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">v</font>) ) &amp; (  not <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">v</font>] implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">v</font>) ) )
 <b>from </b><i><a class="ref" href="pre_circ.html#S2">PRE_CIRC:sch 2</a>();<br/></i>
<a NAME="E3:17_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:17_1_1"/><i><font color="Green">E44</font></i>: 
 <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:17_1_1"/><i><font color="Green">E45</font></i>: 
 <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79">XBOOLE_1:79</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="17_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:17_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font>
 ;<br/><b>then reconsider </b><font color="Maroon">v</font> = <font color="Maroon">x</font> as   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font> <b>by </b><i><a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:17_1_1_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> or  not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E3:17_1_1_1_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="circuit1.html#T5">CIRCUIT1:5</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="17_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:17_1_1_1_1_2"/><i><font color="Green">E47</font></i>: 
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 ;<br/><div class="add"><a NAME="E2:17_1_1_1_1_2"/><b>then </b><i><font color="Green">E48</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">SCS</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E3:17_1_1_1_1_2"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">IIG</font>
 ;<br/><a NAME="E4:17_1_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T45">XBOOLE_1:45</a></i>;<br/><a NAME="E5:17_1_1_1_1_2"/><b>then </b>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E6:17_1_1_1_1_2"/><b>then </b>
 <a href="msualg_1.html#K2">the_result_sort_of</a> <span class="p1">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">v</font>
 <b>by </b><i><a class="ref" href="msafree2.html#D7">MSAFREE2:def 7</a></i>;<br/><b>hence </b><a NAME="E7:17_1_1_1_1_2"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="msualg_1.html#U4">Sorts</a> of <font color="Maroon">SCS</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, <a class="ref" href="circuit1.html#T9">CIRCUIT1:9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="circuit1.html#NM3">State</a> of <font color="Maroon">SCS</font> <b>by </b><i>, , <a class="ref" href="card_3.html#D5">CARD_3:def 5</a></i>;<br/>
<b>take </b>
<font color="Maroon">f</font>
;<br/>

<b>let </b><font color="Maroon">v</font> be   <a href="msafree2.html#NM1">Vertex</a> of <font color="Maroon">IIG</font>;<br/>

<b>thus </b><a NAME="E8:17_1_1"/>
( <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font> implies <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> )
 <b>by </b><i/>;<br/>

<b>assume </b><a NAME="E9:17_1_1"/>
<font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K4">InnerVertices</a> <font color="Maroon">IIG</font>
 ;<br/>

<a NAME="E10:17_1_1"/><b>then </b>
not <font color="Maroon">v</font> <a href="hidden.html#R2">in</a>  <a href="msafree2.html#K2">InputVertices</a> <font color="Maroon">IIG</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#T3">XBOOLE_0:3</a></i>;<br/>
<b>hence </b><a NAME="E11:17_1_1"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="msualg_1.html#K5">Den</a> <span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span>,<font color="Maroon">SCS</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="msafree2.html#K5">action_at</a> <font color="Maroon">v</font></span>)</span> <a href="circuit1.html#K3">depends_on_in</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i/>;<br/>


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