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<b>let </b><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">c</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K3" title="TWOSCOMP:func.3">and2a</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K3" title="TWOSCOMP:func.3">and2a</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> implies for <font color="Olive" title="b1">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K9" title="FSCIRC_1:func.9">BitSubtracterWithBorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,2 is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> )</span><br/>

<b>assume </b><a NAME="E1:32"/><i><font color="Green" title="E25">A1</font></i>: 
( <font color="Maroon" title="c1">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K2" title="TWOSCOMP:func.2">and2</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c2">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c3">c</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K3" title="TWOSCOMP:func.3">and2a</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="twoscomp.html#K3" title="TWOSCOMP:func.3">and2a</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> &amp; <font color="Maroon" title="c3">c</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <span class="p1"><a href="tarski.html#K4" title="TARSKI:func.4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6" title="FACIRC_1:func.6">&lt;*</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font></span><a href="facirc_1.html#K6" title="FACIRC_1:func.6">*&gt;</a></span>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span><a href="tarski.html#K4" title="TARSKI:func.4">]</a></span> )
 ; <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">s</font> being  <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K9" title="FSCIRC_1:func.9">BitSubtracterWithBorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> holds   <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Olive" title="b1">s</font>,2 is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a></span><br/>

<b>set </b><font color="Maroon" title="c4">S</font> =  <a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c5">S1</font> =  <a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> ;<br/>
<b>set </b><font color="Maroon" title="c6">S2</font> =  <a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c7">A</font> =  <a href="fscirc_1.html#K9" title="FSCIRC_1:func.9">BitSubtracterWithBorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c8">A1</font> =  <a href="fscirc_1.html#K2" title="FSCIRC_1:func.2">BitSubtracterCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>set </b><font color="Maroon" title="c9">A2</font> =  <a href="fscirc_1.html#K7" title="FSCIRC_1:func.7">BorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<b>let </b><font color="Maroon" title="c10">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K9" title="FSCIRC_1:func.9">BitSubtracterWithBorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2 is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a></span><br/>

<b>reconsider </b><font color="Maroon" title="c11">s1</font> = <font color="Maroon" title="c10">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K2" title="FSCIRC_1:func.2">BitSubtracterCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T26" title="FACIRC_1:th.26">FACIRC_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c12">s2</font> = <font color="Maroon" title="c10">s</font> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K7" title="FSCIRC_1:func.7">BorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <b>by </b><i><a class="ref" href="facirc_1.html#T26" title="FACIRC_1:th.26">FACIRC_1:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c10">s</font> as   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="fscirc_1.html#K2" title="FSCIRC_1:func.2">BitSubtracterCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="circcomb.html#K5" title="CIRCCOMB:func.5">+*</a> <span class="p2">(<span class="default"><a href="fscirc_1.html#K7" title="FSCIRC_1:func.7">BorrowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span></span>)</span> ;<br/>
<a NAME="E5:32"/><i><font color="Green" title="E26">A2</font></i>: 
 <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="enumset1.html#K1" title="ENUMSET1:func.1">{</a><span class="default"><font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span><a href="enumset1.html#K1" title="ENUMSET1:func.1">}</a></span>
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="fscirc_2.html#T15" target="_self" title="FSCIRC_2:th.15">Th15</a></i>;<br/>
<a NAME="E6:32"/>
(  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="xboole_1.html#T79" title="XBOOLE_1:th.79">XBOOLE_1:79</a></i>;<br/>
<a NAME="E7:32"/><b>then </b>
(  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> &amp;  <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a>  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="txt" href="fscirc_2.html#E5:32"><i><font color="Green" title="E26">A2</font></i></a>, <a class="ref" href="facirc_1.html#T57" title="FACIRC_1:th.57">FACIRC_1:57</a></i>;<br/>
<a NAME="E8:32"/><b>then </b><i><font color="Green" title="E27">A3</font></i>: 
(  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c13">t</font>,2</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> &amp;  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c13">t</font>,3</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> &amp;  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c13">t</font>,2</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> &amp;  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c13">t</font>,3</span>)</span> <a href="relat_1.html#K7" title="RELAT_1:func.7">|</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#T30" title="FACIRC_1:th.30">FACIRC_1:30</a>, <a class="ref" href="facirc_1.html#T31" title="FACIRC_1:th.31">FACIRC_1:31</a></i>;<br/>
<a NAME="E9:32"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2 is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="facirc_1.html#T63" title="FACIRC_1:th.63">FACIRC_1:63</a></i>;<br/>
<b>then </b><i><font color="Green" title="E28">A4</font></i>:  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2 = 
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2</span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a></i>
<br/>.= 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></i>
;<br/>
<a NAME="E11:32"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2 is <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a>
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E1:32"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="fscirc_2.html#T23" target="_self" title="FSCIRC_2:th.23">Th23</a></i>;<br/>
<b>then </b><i><font color="Green" title="E29">A5</font></i>:  <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2 = 
 <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2</span>)</span>
<b>by </b><i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a></i>
<br/>.= 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></i>
;<br/>
<a NAME="E13:32"/><i><font color="Green" title="E30">A6</font></i>: 
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span>
 
<b>by </b><i><a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</a></i>;<br/>
<a NAME="E14:32"/><i><font color="Green" title="E31">A7</font></i>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,3</span>)</span> <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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2</span>)</span> <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 <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> &amp;  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2</span>)</span> <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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E15:32"/><i><font color="Green" title="E32">A8</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> <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 <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> <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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="32_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c14">a</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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> implies <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> )</span><br/><b>assume </b><a NAME="E1:32_3"/>
<font color="Maroon" title="c14">a</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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K8" title="FSCIRC_1:func.8">BitSubtracterWithBorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font></span><br/><a NAME="E2:32_3"/><b>then </b>
( <font color="Maroon" title="c14">a</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 <span class="p1">(<span class="default"><a href="facirc_1.html#K12" title="FACIRC_1:func.12">2GatesCircStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>,<a href="facirc_1.html#K1" title="FACIRC_1:func.1">'xor'</a> </span>)</span> or <font color="Maroon" title="c14">a</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 <span class="p1">(<span class="default"><a href="fscirc_1.html#K4" title="FSCIRC_1:func.4">BorrowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> )
 <b>by </b><i><a class="txt" href="fscirc_2.html#E15:32"><i><font color="Green" title="E32">A8</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:32_3"/><b>then </b>
( ( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c11">s1</font>,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> ) or ( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c12">s2</font>,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> ) )
 <b>by </b><i><a class="txt" href="fscirc_2.html#E8:32"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="fscirc_2.html#E10:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fscirc_2.html#E12:32"><i><font color="Green" title="E29">A5</font></i></a>, <a class="txt" href="fscirc_2.html#E14:32"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></i>;<br/><b>hence </b><a NAME="E4:32_3"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font>
 <b>by </b><i><a class="txt" href="fscirc_2.html#E10:32"><i><font color="Green" title="E28">A4</font></i></a>, <a class="txt" href="fscirc_2.html#E12:32"><i><font color="Green" title="E29">A5</font></i></a>, <a class="ref" href="facirc_1.html#T12" title="FACIRC_1:th.12">FACIRC_1:12</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>
<b>hence </b><a NAME="E17:32"/>
 <a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="circuit2.html#K6" title="CIRCUIT2:func.6">Following</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" title="FACIRC_1:func.9">Following</a> <font color="Maroon" title="c10">s</font>,2</span>)</span>
 <b>by </b><i><a class="txt" href="fscirc_2.html#E13:32"><i><font color="Green" title="E30">A6</font></i></a>, <a class="txt" href="fscirc_2.html#E14:32"><i><font color="Green" title="E31">A7</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</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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