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

<b>set </b><font color="Maroon">f</font> =  <a href="facirc_1.html#K1" target="_self">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S</font> =  <a href="facirc_1.html#K23" target="_self">BitAdderWithOverflowStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">S1</font> =  <a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> ;<br/>
<b>set </b><font color="Maroon">S2</font> =  <a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A</font> =  <a href="facirc_1.html#K24" target="_self">BitAdderWithOverflowCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A1</font> =  <a href="facirc_1.html#K17" target="_self">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>;<br/>
<b>set </b><font color="Maroon">A2</font> =  <a href="facirc_1.html#K22" target="_self">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</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="facirc_1.html#K24" target="_self">BitAdderWithOverflowCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>;<br/><b>let </b><font color="Maroon">a1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> , <font color="Maroon">a3</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="circcomb.html#K10">BOOLEAN</a> ;<br/>



<b>assume </b><a NAME="E1:136"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">a1</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; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 ;<br/>

<a NAME="E2:136"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">x</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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp; <font color="Maroon">y</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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp; <font color="Maroon">c</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="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E3:136"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x</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="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">y</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="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <font color="Maroon">c</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="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> )
 
<b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s1</font> = <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K17" target="_self">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">s2</font> = <font color="Maroon">s</font> <a href="relat_1.html#K7">|</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K22" target="_self">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">t</font> = <font color="Maroon">s</font> as   <a href="circuit1.html#NM3">State</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="facirc_1.html#K17" target="_self">BitAdderCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="circcomb.html#K4">+*</a> <span class="p2">(<span class="default"><a href="facirc_1.html#K22" target="_self">MajorityCirc</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span></span>)</span> ;<br/>
<a NAME="E7:136"/>
( not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> is <a href="facirc_1.html#V2" target="_self">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> is   <a href="relat_1.html#NM1">Relation</a> &amp; not  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> is <a href="facirc_1.html#V2" target="_self">with_pair</a> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> is   <a href="relat_1.html#NM1">Relation</a> )
 
<b>by </b><i>, <a class="ref" href="facirc_1.html#T5" target="_self">Th5</a>, <a class="ref" href="facirc_1.html#T9" target="_self">Th9</a>, </i>;<br/>
<a NAME="E8:136"/><b>then </b><i><font color="Green">E55</font></i>: 
(  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp;  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="xboole_0.html#R1">misses</a>  <a href="msafree2.html#K2">InputVertices</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> )
 
<b>by </b><i/>;<br/>
<a NAME="E9:136"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s1</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K12" target="_self">2GatesCircStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E10:136"/><b>then </b><i><font color="Green">E56</font></i>: 
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#D4" target="_self">Def4</a>, <a class="ref" href="facirc_1.html#D5" target="_self">Def5</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E11:136"/>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s1</font>,2</span>)</span> <a href="facirc_1.html#K15" target="_self">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s1</font>,2</span>)</span> <a href="facirc_1.html#K15" target="_self">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K13" target="_self">2GatesCircOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font>,<a href="facirc_1.html#K1" target="_self">'xor'</a> </span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a3</font> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#D7" target="_self">Def7</a>, , , <a class="ref" href="facirc_1.html#T8" target="_self">Th8</a></i>;<br/>
<b>hence </b><a NAME="E12:136"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K16" target="_self">BitAdderOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K2">'xor'</a> <font color="Maroon">a3</font>
 ;<br/>

<a NAME="E13:136"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">s2</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="facirc_1.html#K19" target="_self">MajorityStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="circuit1.html#T4">CIRCUIT1:4</a></i>;<br/>
<a NAME="E14:136"/><b>then </b>
( <font color="Maroon">a1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">a2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">y</font> &amp; <font color="Maroon">a3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">s2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c</font> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#D4" target="_self">Def4</a>, <a class="ref" href="facirc_1.html#D6" target="_self">Def6</a>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/>
<a NAME="E15:136"/><b>then </b>
( <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">t</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21" target="_self">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,2</span>)</span> <a href="facirc_1.html#K15" target="_self">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21" target="_self">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s2</font>,2</span>)</span> <a href="facirc_1.html#K15" target="_self">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21" target="_self">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font></span>)</span> )
 
<b>by </b><i><a class="ref" href="facirc_1.html#D7" target="_self">Def7</a>, , </i>;<br/>
<b>hence </b><a NAME="E16:136"/>
<span class="p1">(<span class="default"><a href="facirc_1.html#K9" target="_self">Following</a> <font color="Maroon">s</font>,2</span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K21" target="_self">MajorityOutput</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>,<font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a2</font></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p2">(<span class="default"><font color="Maroon">a2</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a3</font></span>)</span></span>)</span> <a href="binarith.html#K1">'or'</a> <span class="p1">(<span class="default"><font color="Maroon">a3</font> <a href="margrel1.html#K10">'&amp;'</a> <font color="Maroon">a1</font></span>)</span>
 ;<br/>


</div>
