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<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">x</font>, <font color="Maroon" title="c3">y</font> be   <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ] <b>means </b> <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default">$1 <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>
<a NAME="E1:15"/>
<a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K6" title="CIRCCOMB:func.6">1GateCircStr</a> <a href="facirc_2.html#K3" title="FACIRC_2:func.3">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="finseq_2.html#K4" title="FINSEQ_2:func.4">-tuples_on</a> <a href="circcomb.html#K9" title="CIRCCOMB:func.9">BOOLEAN</a> </span>)</span> <a href="margrel1.html#K1" title="MARGREL1:func.1">--&gt;</a> <a href="margrel1.html#K8" title="MARGREL1:func.8">TRUE</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="fscirc_2.html#T2" target="_self" title="FSCIRC_2:th.2">Th2</a></i>;<br/>
<a NAME="E2:15"/><b>then </b><i><font color="Green" title="E11">A1</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[ <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ]
 
<b>by </b><i><a class="ref" href="facirc_1.html#T38" title="FACIRC_1:th.38">FACIRC_1:38</a></i>;<br/>
<div><i><font color="Green" title="E12">A2</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" title="c4">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E1:15_1"/><i><font color="Green" title="E13">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c4">i</font>]
 ;<br/><a NAME="E2:15_1"/><i><font color="Green" title="E14">A4</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> <a href="circcomb.html#K3" title="CIRCCOMB:func.3">+*</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="fscirc_2.html#T7" target="_self" title="FSCIRC_2:th.7">Th7</a></i>;<br/><a NAME="E3:15_1"/>
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c4">i</font> <a href="fscirc_2.html#K3" title="FSCIRC_2:func.3">-BitBorrowOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 <b>by </b><i><a class="ref" href="fscirc_1.html#T22" title="FSCIRC_1:th.22">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" title="c4">i</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1]
 <b>by </b><i><a class="txt" href="fscirc_2.html#E1:15_1"><i><font color="Green" title="E13">A3</font></i></a>, <a class="txt" href="fscirc_2.html#E2:15_1"><i><font color="Green" title="E14">A4</font></i></a>, <a class="ref" href="facirc_1.html#T3" title="FACIRC_1:th.3">FACIRC_1:3</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:15"/>
for <font color="Olive" title="b1">i</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1" title="NAT_1:sch.1">NAT_1:sch 1</a>(<a class="txt" href="fscirc_2.html#E2:15"><i><font color="Green" title="E11">A1</font></i></a>, <a class="txt" href="fscirc_2.html#E3:15"><i><font color="Green" title="E12">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:15"/>
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">n</font> <a href="fscirc_2.html#K1" title="FSCIRC_2:func.1">-BitSubtracterStr</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 ;<br/>


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