<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>set </b><font color="Maroon">c</font> = <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>;<br/>
<b>let </b><font color="Maroon">p</font> be    <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">n</font>, <font color="Maroon">q</font> be    <a href="circcomb.html#M1">FinSeqLen</a> of <font color="Maroon">n</font>;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">p2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q1</font> be   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">q2</font> be   <a href="finseq_1.html#NM1">FinSequence</a>;<br/>



<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ) <b>-&gt; </b>   <a href="subset_1.html#M2">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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <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>( 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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <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>3</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="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <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>consider </b><font color="Maroon">f1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h1</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:8"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E3:8"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">f1</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">g1</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">h1</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="E4:8"/><i><font color="Green">E45</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">f1</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">g1</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">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">f1</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="Maroon">H<sub>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">z</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">g1</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="Maroon">H<sub>3</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">z</font>,<font color="Olive">n</font>) &amp; <font color="Maroon">h1</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="Maroon">H<sub>1</sub></font>(<font color="Olive">z</font>,<font color="Olive">n</font>) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">f2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">g2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">h2</font> being    <a href="pboole.html#M1">ManySortedSet</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:8"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>and </b><br/><a NAME="E7:8"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f2</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">g2</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">h2</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="E8:8"/><i><font color="Green">E56</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">f2</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">g2</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">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> holds <br/>( <font color="Maroon">f2</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="fscirc_1.html#K8">BitSubtracterWithBorrowStr</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <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">g2</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="fscirc_1.html#K9">BitSubtracterWithBorrowCirc</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <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"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <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">h2</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="fscirc_1.html#K6">BorrowOutput</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <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"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <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/>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>( <font color="Maroon">a<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies ( <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> ) );<br/>
<a NAME="E9:8"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:8"/><i><font color="Green">E58</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="8_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:8_1"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font> implies ( <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> ) )
 <b>and </b><br/><a NAME="E2:8_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n</font>
 ;<br/>

<a NAME="E3:8_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="circcomb.html#D12">CIRCCOMB:def 12</a></i>;<br/>
<a NAME="E4:8_1"/><b>then </b><i><font color="Green">E62</font></i>: 
(  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> &amp;  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">q</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E5:8_1"/>
0 <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/>
<a NAME="E6:8_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E7:8_1"/><b>then </b><i><font color="Green">E63</font></i>: 
( <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">q</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#D7">FINSEQ_1:def 7</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>verum;<br/>
<a NAME="E8:8_1"/><i><font color="Green">E64</font></i>: 
 ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <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> ex <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,0] )
 
<b>by </b><i/>;<br/>
<a NAME="E9:8_1"/><i><font color="Green">E68</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">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</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">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">H<sub>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">H<sub>3</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>,<font color="Olive">n</font>),<font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1]
 
;<br/>
<a NAME="E10:8_1"/><i><font color="Green">E69</font></i>: 
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> <br/> for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Olive">z</font>,<font color="Olive">n</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>2</sub></font>(<font color="Olive">S</font>,<font color="Olive">z</font>,<font color="Olive">n</font>)
 
;<br/>
<a NAME="E11:8_1"/>
for <font color="Olive">n</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   ex <font color="Olive">S</font> being non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a>  ex <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> st <br/>( <font color="Olive">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</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">g1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">S</font>,<font color="Olive">A</font>,<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>,<font color="Olive">n</font>] )
 
<b>from </b><i><a class="ref" href="circcmb2.html#S13">CIRCCMB2:sch 13</a>(, , , );<br/></i>
<b>then consider </b><font color="Maroon">S</font> being  non <a href="struct_0.html#V3">empty</a>  <a href="msualg_1.html#L1">ManySortedSign</a> , <font color="Maroon">A</font> being  <a href="msualg_1.html#V5">non-empty</a>  <a href="msualg_1.html#L3">MSAlgebra</a> of <font color="Maroon">S</font><b> such that </b><br/><a NAME="E13:8_1"/><i><font color="Green">E84</font></i>: 
( <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font> )
 ;<br/>
<b>thus </b><font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
 <a href="fscirc_1.html#K6">BorrowOutput</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
;<br/>

<b>thus </b><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">S</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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
;<br/>

<b>thus </b><font color="Maroon">g1</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> = 
<font color="Maroon">A</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"><span class="p3">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="funct_1.html#K1">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font></span>)</span></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
<br/>.= 
<font color="Maroon">g2</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i>, , , , <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E11:8"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">i</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>hence </b><a NAME="E12:8"/>
( <font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> &amp; <font color="Maroon">n</font> <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="fscirc_2.html#K2" target="_self">-BitSubtracterCirc</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> )
 <b>by </b><i>, </i>;<br/>

<a NAME="E13:8"/>
( <font color="Maroon">n</font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">n</font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E14:8"/>
<font color="Maroon">n</font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p1</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> <a href="fscirc_2.html#K3" target="_self">-BitBorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">p2</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">q</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">q2</font></span>)</span>
 <b>by </b><i/>;<br/>


</div>
