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<b>consider </b><font color="Maroon" title="c5">i</font> being   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a><b> such that </b><br/><a NAME="E2:34_1_1"/><i><font color="Green" title="E24">A2</font></i>: 
<font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <font color="Maroon" title="c5">i</font>
 <b>by </b><i><a class="txt" href="#E1:34"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="nat_1.html#T10" title="NAT_1:th.10">NAT_1:10</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c6">i</font> = <font color="Maroon" title="c5">i</font> as   <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> ;<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> non <a href="struct_0.html#V11" title="STRUCT_0:attr.11">void</a> <a href="msualg_1.html#V1" title="MSUALG_1:attr.1">strict</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="c3">x</font>,<font color="Maroon" title="c4">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="c3">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="c4">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="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>;<br/>
<b>set </b><font color="Maroon" title="c7">o</font> =  <a href="gfacirc1.html#K24" title="GFACIRC1:func.24">GFA1AdderOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span>;<br/>
<a NAME="E4:34_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">k</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c2">n</font>)
 
<b>by </b><i><a class="txt" href="#E1:34"><i><font color="Green" title="E23">A1</font></i></a>, <a class="ref" href="gfacirc2.html#T22" target="_self" title="GFACIRC2:th.22">Th22</a></i>;<br/>
<a NAME="E5:34_1_1"/><i><font color="Green" title="E26">A4</font></i>: 
 <a href="gfacirc1.html#K24" title="GFACIRC1:func.24">GFA1AdderOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>)
 
<b>by </b><i><a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="gfacirc1.html#T82" title="GFACIRC1:th.82">GFACIRC1:82</a></i>;<br/>
<a NAME="E6:34_1_1"/><i><font color="Green" title="E27">A5</font></i>: 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="circcomb.html#K4" title="CIRCCOMB:func.4">+*</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>)
 
<b>by </b><i><a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="gfacirc2.html#T21" target="_self" title="GFACIRC2:th.21">Th21</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">o</font> =  <a href="gfacirc1.html#K24" title="GFACIRC1:func.24">GFA1AdderOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c6">i</font> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) <b>by </b><i><a class="txt" href="#E5:34_1_1"><i><font color="Green" title="E26">A4</font></i></a></i>;<br/>
<a NAME="E8:34_1_1"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">k</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon" title="c6">i</font>) <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c6">i</font>)
 
<b>by </b><i><a class="txt" href="#E6:34_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></i>;<br/>
<a NAME="E9:34_1_1"/><b>then </b>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">k</font>)
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:34_1_1"/><b>then </b>
<font color="Maroon" title="c8">o</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c1">k</font>)
 
<b>by </b><i><a class="txt" href="#E5:34_1_1"><i><font color="Green" title="E26">A4</font></i></a>, <a class="txt" href="#E6:34_1_1"><i><font color="Green" title="E27">A5</font></i></a>, <a class="ref" href="circcomb.html#T19" title="CIRCCOMB:th.19">CIRCCOMB:19</a></i>;<br/>
<b>hence </b><a NAME="E11:34_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">n</font> <a href="gfacirc2.html#K5" title="GFACIRC2:func.5">-BitGFA1Str</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> ex <font color="Olive" title="b2">i</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a> st <br/>( <font color="Maroon" title="c1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">i</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 &amp; <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="gfacirc1.html#K24" title="GFACIRC1:func.24">GFA1AdderOutput</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c4">y</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c1">k</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive" title="b2">i</font> <a href="gfacirc2.html#K7" title="GFACIRC2:func.7">-BitGFA1CarryOutput</a> <font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span>)</span> )
 <b>by </b><i><a class="txt" href="#E2:34_1_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="#E4:34_1_1"><i><font color="Green" title="E25">A3</font></i></a></i>; <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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