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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>5</sub></font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:5"/><i><font color="Green">E6</font></i>: 
( 0 <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; 0 <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 )
 <b>and </b><br/><a NAME="E3:5"/><i><font color="Green">E7</font></i>: 
<font color="Maroon">c<sub>3</sub></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">&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#K8">TRUE</a> </span>)</span>
 <b>and </b><br/><a NAME="E4:5"/><i><font color="Green">E8</font></i>: 
<font color="Maroon">c<sub>4</sub></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">&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#K8">TRUE</a> </span>)</span>
 <b>and </b><br/><a NAME="E5:5"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><a href="facirc_2.html#K3">&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#K8">TRUE</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>and </b><br/><a NAME="E6:5"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> <br/>for <font color="Olive">b<sub>3</sub></font> being <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Olive">b<sub>2</sub></font><br/>for <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="circcomb.html#K3">+*</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">b<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> <a href="circcomb.html#K4">+*</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K9">BitSubtracterWithBorrowCirc</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">b<sub>4</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="fscirc_1.html#K6">BorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Olive">b<sub>4</sub></font> )
 <b>by </b><i><a class="ref" href="fscirc_2.html#D2" target="_self">Def2</a></i>;<br/>
<b>thus </b><a NAME="E7:5"/>
0 <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K7">1GateCircStr</a> <a href="facirc_2.html#K3">&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#K8">TRUE</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fscirc_2.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fscirc_2.html#E3:5"><i><font color="Green">E7</font></i></a></i>;<br/>

<b>thus </b><a NAME="E8:5"/>
0 <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="circcomb.html#K9">1GateCircuit</a> <a href="facirc_2.html#K3">&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#K8">TRUE</a> </span>)</span>
 <b>by </b><i><a class="txt" href="fscirc_2.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fscirc_2.html#E4:5"><i><font color="Green">E8</font></i></a></i>;<br/>

<a NAME="E9:5"/>
 <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default">0 <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2"><a href="tarski.html#K4">[</a><span class="default"><a href="facirc_2.html#K3">&lt;*&gt;</a> ,<span class="p3">(<span class="default"><span class="p4">(<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#K8">TRUE</a> </span>)</span></span><a href="tarski.html#K4">]</a></span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E2:5"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fscirc_2.html#E3:5"><i><font color="Green">E7</font></i></a>, <a class="ref" href="circcomb.html#T49">CIRCCOMB:49</a></i>;<br/>
<b>hence </b><a NAME="E10:5"/>
0 <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></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">&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#K8">TRUE</a> </span>)</span></span><a href="tarski.html#K4">]</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>


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