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<b>let </b><font color="Maroon" title="c1">n</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being  <a href="finseq_1.html#NM1" title="FINSEQ_1:NM.1">FinSequence</a> holds   <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="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> <font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font></span>)</span> is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a></span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <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="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</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></span><br/>



<b>set </b><font color="Maroon" title="c4">S0</font> =  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">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#K10" title="CIRCCOMB:func.10">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>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b> non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</a> non <a href="msualg_1.html#V2" title="MSUALG_1:attr.2">void</a> <a href="circcomb.html#V1" title="CIRCCOMB:attr.1">unsplit</a> <a href="circcomb.html#V2" title="CIRCCOMB:attr.2">gate`1=arity</a> <a href="circcomb.html#V3" title="CIRCCOMB:attr.3">gate`2isBoolean</a>  <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  = $1 <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="msualg_1.html#L1" title="MSUALG_1:struct.1">ManySortedSign</a>  =  <a href="gfacirc1.html#K25" title="GFACIRC1:func.25">BitGFA1Str</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c3">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">$1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default">$1 <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> <font color="Maroon" title="c2">x</font>,<font color="Maroon" title="c3">y</font></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>] <b>means </b> <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>($1) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>;<br/>
<a NAME="E1:36"/>
<font color="Maroon">H<sub>1</sub></font>( <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circcomb.html#K7" title="CIRCCOMB:func.7">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#K10" title="CIRCCOMB:func.10">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="gfacirc2.html#T16" target="_self" title="GFACIRC2:th.16">Th16</a></i>;<br/>
<a NAME="E2:36"/><b>then </b><i><font color="Green" title="E24">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="E25">A2</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">i</font> be   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )</span><br/><b>assume </b><a NAME="E1:36_1"/><i><font color="Green" title="E26">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font>]
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]</span><br/><a NAME="E2:36_1"/><i><font color="Green" title="E27">A4</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c5">i</font>) <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>)
 <b>by </b><i><a class="ref" href="gfacirc2.html#T21" target="_self" title="GFACIRC2:th.21">Th21</a></i>;<br/><a NAME="E3:36_1"/>
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c5">i</font>) is   <a href="relat_1.html#NM1" title="RELAT_1:NM.1">Relation</a>
 <b>by </b><i><a class="ref" href="gfacirc1.html#T77" title="GFACIRC1:th.77">GFACIRC1:77</a></i>;<br/><b>hence </b><a NAME="E4:36_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="gfacirc2.html#E1:36_1"><i><font color="Green" title="E26">A3</font></i></a>, <a class="txt" href="gfacirc2.html#E2:36_1"><i><font color="Green" title="E27">A4</font></i></a>, <a class="ref" href="facirc_1.html#T3" title="FACIRC_1:th.3">FACIRC_1:3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E4:36"/>
for <font color="Olive" title="b1">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">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#S2" title="NAT_1:sch.2">NAT_1:sch 2</a>(<a class="txt" href="gfacirc2.html#E2:36"><i><font color="Green" title="E24">A1</font></i></a>, <a class="txt" href="gfacirc2.html#E3:36"><i><font color="Green" title="E25">A2</font></i></a>);<br/></i>
<b>hence </b><a NAME="E5:36"/>
 <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="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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