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<b>let </b><font color="Maroon">a</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">b</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:18"/><i><font color="Green">E25</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 ;<br/>

<b>consider </b><font color="Maroon">f</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E3:18"/><i><font color="Green">E33</font></i>: 
1 <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1
 <b>and </b><br/><a NAME="E4:18"/><i><font color="Green">E37</font></i>: 
1 <a href="facirc_2.html#K5" target="_self">-BitAdderCirc</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1
 <b>and </b><br/><a NAME="E5:18"/><i><font color="Green">E38</font></i>: 
( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K7">1GateCircStr</a> <a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K9">1GateCircuit</a> <a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c</font> )
 <b>and </b><br/><a NAME="E6:18"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/> for <font color="Olive">A</font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">S</font><br/> for <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">z</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">S</font> <a href="circcomb.html#K3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K23">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">z</font></span>)</span> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K24">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">z</font></span>)</span> &amp; <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K21">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">z</font> )
 <b>by </b><i>, </i>;<br/>
<a NAME="E7:18"/>
( <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#K23">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, , , , <a class="ref" href="facirc_1.html#T87">FACIRC_1:87</a></i>;<br/>
<b>hence </b><a NAME="E8:18"/>
( 1 <a href="facirc_2.html#K4" target="_self">-BitAdderStr</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K7">1GateCircStr</a> <a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K3">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K23">BitAdderWithOverflowStr</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K5" target="_self">-BitAdderCirc</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="circcomb.html#K9">1GateCircuit</a> <a href="facirc_2.html#K3" target="_self">&lt;*&gt;</a> ,<span class="p2">(<span class="default"><span class="p3">(<span class="default">0 <a href="finseq_2.html#K4">-tuples_on</a> <a href="circcomb.html#K10">BOOLEAN</a> </span>)</span> <a href="circcomb.html#K5">--&gt;</a> <a href="margrel1.html#K7">FALSE</a> </span>)</span></span>)</span> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="facirc_1.html#K24">BitAdderWithOverflowCirc</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c</font></span>)</span> &amp; 1 <a href="facirc_2.html#K6" target="_self">-BitMajorityOutput</a> <font color="Maroon">a</font>,<font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="facirc_1.html#K21">MajorityOutput</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>,<font color="Maroon">c</font> )
 <b>by </b><i>, , , </i>;<br/>


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