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<span class="kw">let </span><font color="Maroon" title="c1">x</font>, <font color="Maroon" title="c2">y</font>, <font color="Maroon" title="c3">c</font> be  non  <a href="facirc_1.html#V1" title="FACIRC_1:attr.1">pair</a>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</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#K5" title="FACIRC_1:func.5">Following</a> <font color="Olive" title="b1">s</font>,2 is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c4">S1</font> =  <a href="facirc_1.html#K8" title="FACIRC_1:func.8">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/>
<span class="kw">set </span><font color="Maroon" title="c5">S2</font> =  <a href="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c6">A</font> =  <a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c7">A1</font> =  <a href="facirc_1.html#K13" title="FACIRC_1:func.13">BitAdderCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<span class="kw">set </span><font color="Maroon" title="c8">A2</font> =  <a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityCirc</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<span class="kw">let </span><font color="Maroon" title="c9">s</font> be   <a href="circuit1.html#NM3" title="CIRCUIT1:NM.3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K20" title="FACIRC_1:func.20">BitAdderWithOverflowCirc</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide">  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">s</font>,2 is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> </span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">s2</font> = <font color="Maroon" title="c9">s</font> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K15" title="FACIRC_1:func.15">MajorityStr</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="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityCirc</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 class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T26" target="_self" title="FACIRC_1:th.26">Th26</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c11">t</font> = <font color="Maroon" title="c9">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="facirc_1.html#K13" title="FACIRC_1:func.13">BitAdderCirc</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#K3" title="CIRCCOMB:func.3">+*</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K18" title="FACIRC_1:func.18">MajorityCirc</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/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">s1</font> = <font color="Maroon" title="c9">s</font> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K8" title="FACIRC_1:func.8">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="facirc_1.html#K13" title="FACIRC_1:func.13">BitAdderCirc</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 class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T26" target="_self" title="FACIRC_1:th.26">Th26</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c13">S</font> =  <a href="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font>;<br/>
<a NAME="E4:131"/><span class="lab"><font color="Green" title="E69">A1</font></span>: 
 <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></span>;<br/>
<a NAME="E5:131"/><span class="lab"><font color="Green" title="E70">A2</font></span>: 
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#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</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#K8" title="FACIRC_1:func.8">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="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="circcomb.html#D2" title="CIRCCOMB:def.2">CIRCCOMB:def 2</a></span>;<br/>
<a NAME="E6:131"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">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> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T59" target="_self" title="FACIRC_1:th.59">Th59</a></span>;<br/>
<a NAME="E7:131"/><span class="kw">then </span>
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</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#K8" title="FACIRC_1:func.8">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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T5" target="_self" title="FACIRC_1:th.5">Th5</a>, <a class="ref" href="facirc_1.html#T67" target="_self" title="FACIRC_1:th.67">Th67</a></span>;<br/>
<a NAME="E8:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E71">A3</font></span>: 
(  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s1</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c11">t</font>,2</span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K8" title="FACIRC_1:func.8">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s1</font>,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c11">t</font>,3</span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K8" title="FACIRC_1:func.8">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> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T30" target="_self" title="FACIRC_1:th.30">Th30</a></span>;<br/>
<a NAME="E9:131"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s1</font>,2 is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T63" target="_self" title="FACIRC_1:th.63">Th63</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E72">A4</font></span>:  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s1</font>,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a></span>
<br/>.= 
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">s1</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a></span>
;<br/>
<a NAME="E11:131"/>
not  <a href="msafree2.html#K2" title="MSAFREE2:func.2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</a> <font color="Maroon" title="c1">x</font>,<font color="Maroon" title="c2">y</font>,<font color="Maroon" title="c3">c</font></span>)</span> is  <a href="facirc_1.html#V2" title="FACIRC_1:attr.2">with_pair</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T68" target="_self" title="FACIRC_1:th.68">Th68</a></span>;<br/>
<a NAME="E12:131"/><span class="kw">then </span>
 <a href="msafree2.html#K4" title="MSAFREE2:func.4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K8" title="FACIRC_1:func.8">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#K15" title="FACIRC_1:func.15">MajorityStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T5" target="_self" title="FACIRC_1:th.5">Th5</a>, <a class="ref" href="facirc_1.html#T58" target="_self" title="FACIRC_1:th.58">Th58</a></span>;<br/>
<a NAME="E13:131"/><span class="kw">then </span><span class="lab"><font color="Green" title="E73">A5</font></span>: 
(  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c11">t</font>,2</span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K15" title="FACIRC_1:func.15">MajorityStr</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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,3 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c11">t</font>,3</span>)</span> <a href="card_3.html#K11" title="CARD_3:func.11">|</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#K15" title="FACIRC_1:func.15">MajorityStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T31" target="_self" title="FACIRC_1:th.31">Th31</a></span>;<br/>
<a NAME="E14:131"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,2 is  <a href="circuit2.html#V1" title="CIRCUIT2:attr.1">stable</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T84" target="_self" title="FACIRC_1:th.84">Th84</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E74">A6</font></span>:  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,2</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a></span>
<br/>.= 
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,<span class="p1">(<span class="default">2 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1</span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a></span>
;<br/>
<a NAME="E16:131"/><span class="lab"><font color="Green" title="E75">A7</font></span>: 
(  <a href="relat_1.html#NK2" title="RELAT_1:NK.2">dom</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">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#K8" title="FACIRC_1:func.8">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">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="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E76">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="131_3"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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:()"><span class="comment"><font color="Red">::  thesis: </font></span></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="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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/><span class="kw">assume </span><a NAME="E1:131_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="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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="E2:131_3"/><span class="kw">then </span>
( <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#K8" title="FACIRC_1:func.8">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="facirc_1.html#K15" title="FACIRC_1:func.15">MajorityStr</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 class="kw">by</span> <span class="lab"><a class="txt" href="#E5:131"><span class="lab"><font color="Green" title="E70">A2</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E3:131_3"/><span class="kw">then </span>
( ( <span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c12">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c10">s2</font>,3</span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">a</font> ) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:131"><span class="lab"><font color="Green" title="E71">A3</font></span></a>, <a class="txt" href="#E13:131"><span class="lab"><font color="Green" title="E73">A5</font></span></a>, <a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E15:131"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="txt" href="#E16:131"><span class="lab"><font color="Green" title="E75">A7</font></span></a>, <a class="ref" href="funct_1.html#T70" title="FUNCT_1:th.70">FUNCT_1:70</a></span>;<br/><span class="kw">hence </span><a NAME="E4:131_3"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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 class="kw">by</span> <span class="lab"><a class="txt" href="#E10:131"><span class="lab"><font color="Green" title="E72">A4</font></span></a>, <a class="txt" href="#E15:131"><span class="lab"><font color="Green" title="E74">A6</font></span></a>, <a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E18:131"/>
(  <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">s</font>,2</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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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="facirc_1.html#K19" title="FACIRC_1:func.19">BitAdderWithOverflowStr</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 class="kw">by</span> <span class="lab"><a class="ref" href="facirc_1.html#T12" target="_self" title="FACIRC_1:th.12">Th12</a>, <a class="ref" href="circuit1.html#T4" title="CIRCUIT1:th.4">CIRCUIT1:4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E19:131"/>
 <a href="facirc_1.html#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">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#K5" title="FACIRC_1:func.5">Following</a> <font color="Maroon" title="c9">s</font>,2</span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:131"><span class="lab"><font color="Green" title="E69">A1</font></span></a>, <a class="txt" href="#E17:131"><span class="lab"><font color="Green" title="E76">A8</font></span></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></span>; <span class="comment"><font color="Red">:: according to </font></span><a class="ref" href="circuit2.html#D6" title="CIRCUIT2:def.6">CIRCUIT2:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
