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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>let </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> be    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="struct_0.html#M1">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4">BorrowStr</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="Maroon">a<sub>2</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="Maroon">a<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>1</sub></font></span>)</span> =  <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="Maroon">a<sub>2</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="Maroon">a<sub>2</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="struct_0.html#M1">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><a href="fscirc_1.html#K4">BorrowStr</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="Maroon">a<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="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span>)</span> =  <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="Maroon">a<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="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>2</sub></font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>( non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ,   <a href="hidden.html#M1">set</a> ,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="msualg_1.html#L1">ManySortedSign</a>  = <font color="Maroon">a<sub>1</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="Maroon">a<sub>3</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="Maroon">a<sub>3</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span>)</span>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>4</sub></font>( non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">a<sub>1</sub></font>,   <a href="hidden.html#M1">set</a> ,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">a<sub>1</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="Maroon">a<sub>4</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="Maroon">a<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>3</sub></font></span>)</span> = <font color="Maroon">a<sub>2</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="Maroon">a<sub>4</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="Maroon">a<sub>4</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<font color="Maroon">a<sub>3</sub></font></span>)</span>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:4"/><i><font color="Green">E5</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> &amp; <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="E2:4"/><i><font color="Green">E6</font></i>: 
<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="E3:4"/><i><font color="Green">E7</font></i>: 
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="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &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="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</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="E5:4"/><i><font color="Green">E8</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <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>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <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>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>and </b><br/><a NAME="E6:4"/><i><font color="Green">E9</font></i>: 
( <font color="Maroon">c<sub>7</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> &amp; <font color="Maroon">c<sub>8</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> &amp; <font color="Maroon">c<sub>9</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="E7:4"/><i><font color="Green">E10</font></i>: 
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>7</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>8</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>9</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Maroon">c<sub>7</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="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>8</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="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) &amp; <font color="Maroon">c<sub>9</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="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 <b>by </b><i><a class="ref" href="fscirc_2.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:4"/><i><font color="Green">E11</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> ex <font color="Olive">b<sub>2</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>1</sub></font> st <br/>( <font color="Olive">b<sub>1</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; <font color="Olive">b<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>by </b><i><a class="txt" href="fscirc_2.html#E1:4"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E9:4"/><i><font color="Green">E12</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>4</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> <a href="funct_1.html#K1">.</a> 0 )
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="fscirc_2.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="txt" href="fscirc_2.html#E6:4"><i><font color="Green">E9</font></i></a></i>;<br/>
<a NAME="E10:4"/><i><font color="Green">E13</font></i>: 
for <font color="Olive">b<sub>1</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>2</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>1</sub></font><br/>for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a> <br/>for <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>) is  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>4</sub></font>)
 
;<br/>
<a NAME="E11:4"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>9</sub></font> )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S14">CIRCCMB2:sch 14</a>(<a class="txt" href="fscirc_2.html#E8:4"><i><font color="Green">E11</font></i></a>, <a class="txt" href="fscirc_2.html#E9:4"><i><font color="Green">E12</font></i></a>, <a class="txt" href="fscirc_2.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fscirc_2.html#E7:4"><i><font color="Green">E10</font></i></a>, <a class="txt" href="fscirc_2.html#E10:4"><i><font color="Green">E13</font></i></a>);<br/></i>
<b>hence </b><a NAME="E12:4"/>
( <font color="Maroon">c<sub>6</sub></font> <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> <font color="Maroon">c<sub>6</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <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> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="txt" href="fscirc_2.html#E5:4"><i><font color="Green">E8</font></i></a></i>;<br/>

<a NAME="E13:4"/><i><font color="Green">E14</font></i>: 
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="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>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="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) &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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>1</sub></font>) )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S15">CIRCCMB2:sch 15</a>(<a class="txt" href="fscirc_2.html#E1:4"><i><font color="Green">E5</font></i></a>, <a class="txt" href="fscirc_2.html#E3:4"><i><font color="Green">E7</font></i></a>, <a class="txt" href="fscirc_2.html#E10:4"><i><font color="Green">E13</font></i></a>);<br/></i>
<a NAME="E14:4"/><i><font color="Green">E15</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>by </b><i><a class="txt" href="fscirc_2.html#E1:4"><i><font color="Green">E5</font></i></a></i>;<br/>
<a NAME="E15:4"/>
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   <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>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>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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font>)
 
<b>from </b><i><a class="ref" href="circcmb2.html#S3">CIRCCMB2:sch 3</a>(<a class="txt" href="fscirc_2.html#E14:4"><i><font color="Green">E15</font></i></a>, <a class="txt" href="fscirc_2.html#E13:4"><i><font color="Green">E14</font></i></a>);<br/></i>
<a NAME="E16:4"/><b>then </b><i><font color="Green">E16</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <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> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>) ) )
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E2:4"><i><font color="Green">E6</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>10</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="E18:4"/><i><font color="Green">E17</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <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> <font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>and </b><br/><a NAME="E19:4"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>10</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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</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>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> ) )
 <b>by </b><i><a class="ref" href="fscirc_2.html#D3" target="_self">Def3</a></i>;<br/>
<a NAME="E20:4"/><i><font color="Green">E19</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>10</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> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>10</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="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>) ) )
 
<b>by </b><i><a class="txt" href="fscirc_2.html#E19:4"><i><font color="Green">E18</font></i></a>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E21:4"/>
<font color="Maroon">c<sub>5</sub></font> <a href="pboole.html#R6">=</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S11">RECDEF_1:sch 11</a>(<a class="txt" href="fscirc_2.html#E16:4"><i><font color="Green">E16</font></i></a>, <a class="txt" href="fscirc_2.html#E20:4"><i><font color="Green">E19</font></i></a>);<br/></i>
<b>hence </b><a NAME="E22:4"/>
<font color="Maroon">c<sub>6</sub></font> <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> <font color="Maroon">c<sub>5</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>6</sub></font>
 <b>by </b><i><a class="txt" href="fscirc_2.html#E18:4"><i><font color="Green">E17</font></i></a></i>;<br/>


</div>
