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<div class="add">

<b>let </b><font color="Maroon">x</font> be  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> , <font color="Maroon">b</font> be  non <a href="facirc_1.html#V1">pair</a>  <a href="hidden.html#M1">set</a> ;<br/>

<b>set </b><font color="Maroon">p</font> = <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="twoscomp.html#K40" target="_self">BitCompStr</a> <font color="Maroon">x</font>,<font color="Maroon">b</font>;<br/>
<b>let </b><font color="Maroon">s</font> be   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="twoscomp.html#K41" target="_self">BitCompCirc</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>;<br/>

<b>set </b><font color="Maroon">s1</font> =  <a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font>;<br/>
<b>set </b><font color="Maroon">s2</font> =  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span>;<br/>
<a NAME="E1:61"/><i><font color="Green">E70</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="twoscomp.html#K40" target="_self">BitCompStr</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span> &amp;  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="twoscomp.html#K40" target="_self">BitCompStr</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E2:61"/><i><font color="Green">E87</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="twoscomp.html#K40" target="_self">BitCompStr</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K15" target="_self">xor2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3" target="_self">and2a</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span>
 
<b>by </b><i><a class="ref" href="twoscomp.html#D13" target="_self">Def13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="61_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:61_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="twoscomp.html#K40" target="_self">BitCompStr</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>
 ;<br/><a NAME="E2:61_1"/><b>then </b>
( <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="tarski.html#K2">}</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="tarski.html#K2">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K15" target="_self">xor2a</a> </span><a href="tarski.html#K4">]</a></span>,<span class="p2"><a href="tarski.html#K4">[</a><span class="default"><span class="p3"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3" target="_self">and2a</a> </span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K2">}</a></span> )
 <b>by </b><i><a class="ref" href="twoscomp.html#D24" target="_self">Def24</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><a NAME="E3:61_1"/><b>then </b><i><font color="Green">E88</font></i>: 
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K15" target="_self">xor2a</a> </span><a href="tarski.html#K4">]</a></span> or <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3" target="_self">and2a</a> </span><a href="tarski.html#K4">]</a></span> )
 <b>by </b><i><a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/><a NAME="E4:61_1"/><i><font color="Green">E89</font></i>: 
( <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> &amp; <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font> )
 <b>by </b><i><a class="ref" href="twoscomp.html#D18" target="_self">Def18</a></i>;<br/><i><font color="Green">E93</font></i>: <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K15" target="_self">xor2a</a> </span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="twoscomp.html#K36" target="_self">CompOutput</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>

<br/>.= 
<a href="twoscomp.html#K15" target="_self">xor2a</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="facirc_1.html#K6">*&gt;</a></span>
<b>by </b><i><a class="ref" href="twoscomp.html#D18" target="_self">Def18</a></i>
;<br/><i><font color="Green">E101</font></i>: <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3" target="_self">and2a</a> </span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="twoscomp.html#K39" target="_self">IncrementOutput</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>

<br/>.= 
<a href="twoscomp.html#K3" target="_self">and2a</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="facirc_1.html#K6">*&gt;</a></span>
<b>by </b><i><a class="ref" href="twoscomp.html#D21" target="_self">Def21</a></i>
;<br/><i><font color="Green">E102</font></i>: <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K15" target="_self">xor2a</a> </span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="twoscomp.html#K36" target="_self">CompOutput</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>

<br/>.= 
<a href="twoscomp.html#K15" target="_self">xor2a</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="facirc_1.html#K6">*&gt;</a></span>
<b>by </b><i><a class="ref" href="twoscomp.html#D18" target="_self">Def18</a></i>
;<br/><span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><span class="p2"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">b</font></span><a href="facirc_1.html#K6">*&gt;</a></span>,<a href="twoscomp.html#K3" target="_self">and2a</a> </span><a href="tarski.html#K4">]</a></span> = 
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="twoscomp.html#K39" target="_self">IncrementOutput</a> <font color="Maroon">x</font>,<font color="Maroon">b</font></span>)</span>

<br/>.= 
<a href="twoscomp.html#K3" target="_self">and2a</a>  <a href="funct_1.html#K1">.</a> <span class="p1"><a href="facirc_1.html#K6">&lt;*</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">b</font></span>)</span></span><a href="facirc_1.html#K6">*&gt;</a></span>
<b>by </b><i><a class="ref" href="twoscomp.html#D21" target="_self">Def21</a></i>
;<br/><b>hence </b><a NAME="E9:61_1"/>
<span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <span class="p2">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a</font>
 <b>by </b><i><a class="ref" href="twoscomp.html#D25" target="_self">Def25</a>, <a class="ref" href="twoscomp.html#D26" target="_self">Def26</a>, <a class="ref" href="twoscomp.html#D27" target="_self">Def27</a>, <a class="ref" href="twoscomp.html#D28" target="_self">Def28</a>, <a class="ref" href="twoscomp.html#D29" target="_self">Def29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:61"/>
 <a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="circuit2.html#K6">Following</a> <span class="p1">(<span class="default"><a href="circuit2.html#K6">Following</a> <font color="Maroon">s</font></span>)</span>
 <b>by </b><i><a class="ref" href="twoscomp.html#D23" target="_self">Def23</a>, <a class="ref" href="funct_1.html#T9">FUNCT_1:9</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="circuit2.html#D6">CIRCUIT2:def 6</a><br/>


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