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<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is   <a href="relat_1.html#NM1">Relation</a>;<br/>
<a NAME="E1:15"/>
0 <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K7">1GateCircStr</a> <a href="facirc_2.html#K3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K8">TRUE</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="fscirc_2.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green">E11</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i><a class="ref" href="facirc_1.html#T38">FACIRC_1:38</a></i>;<br/>
<div><i><font color="Green">E12</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>]
 ;<br/><a NAME="E2:15_1"/><i><font color="Green">E14</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fscirc_2.html#T7" target="_self">Th7</a></i>;<br/><a NAME="E3:15_1"/>
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span></span>)</span> is   <a href="relat_1.html#NM1">Relation</a>
 <b>by </b><i><a class="ref" href="fscirc_1.html#T22">FSCIRC_1:22</a></i>;<br/><b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="fscirc_2.html#E1:15_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="fscirc_2.html#E2:15_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="facirc_1.html#T3">FACIRC_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:15"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</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="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="fscirc_2.html#E2:15"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fscirc_2.html#E3:15"><i><font color="Green">E12</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:15"/>
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>3</sub></font></span>)</span> is   <a href="relat_1.html#NM1">Relation</a>
 ;<br/>


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