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<b>let </b><font color="Maroon">o1</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>, <font color="Maroon">o2</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="msafree2.html#K4">InnerVertices</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="fscirc_2.html#K1" target="_self">-BitSubtracterStr</a> <font color="Maroon">x</font>,<font color="Maroon">y</font></span>)</span>;<br/>

<b>given </b><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><a NAME="E1:3_1_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">o1</font> <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">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> &amp; ( 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">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>  <a href="fscirc_1.html#K6">BorrowOutput</a> <span class="p1">(<span class="default"><font color="Maroon">x</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">y</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">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>given </b><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><a NAME="E2:3_1_1"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">o2</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> &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> &amp; ( 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">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"><font color="Maroon">x</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">y</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">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> ) )
 ;<br/>

<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ,   <a href="hidden.html#M1">set</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"><font color="Maroon">x</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">y</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">x</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">y</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/>
<a NAME="E3:3_1_1"/><i><font color="Green">E45</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h1</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &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> &amp; ( 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">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">n</font>,<font color="Maroon">h1</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>) ) )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E4:3_1_1"/><i><font color="Green">E46</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <font color="Maroon">h2</font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &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> &amp; ( 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">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> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">n</font>,<font color="Maroon">h2</font> <a href="funct_1.html#K1">.</a> <font color="Olive">n</font>) ) )
 
<b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a></i>;<br/>
<a NAME="E5:3_1_1"/>
<font color="Maroon">h1</font> <a href="pboole.html#R6">=</a> <font color="Maroon">h2</font>
 
<b>from </b><i><a class="ref" href="recdef_1.html#S11">RECDEF_1:sch 11</a>(, );<br/></i>
<b>hence </b><a NAME="E6:3_1_1"/>
<font color="Maroon">o1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">o2</font>
 <b>by </b><i>, </i>;<br/>


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