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

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font>, <font color="Maroon">c<sub>6</sub></font> be   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default"><span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:59_1_2"/><i><font color="Green">E35</font></i>: 
(  <a href="relset_1.html#K4">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_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  ) )
 <b>and </b><br/><a NAME="E2:59_1_2"/><i><font color="Green">E36</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</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> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>3</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>3</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  ) )
 ;<br/>

<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Olive">b<sub>2</sub></font> <a href="subset_1.html#K6">\</a> <span class="p3"><a href="domain_1.html#K6">{</a><span class="default"><span class="p4">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">a<sub>2</sub></font> ) </span>}</span> ;<br/>
<a NAME="E3:59_1_2"/><i><font color="Green">E37</font></i>: 
(  <a href="relset_1.html#K4">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_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#K3">.</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>] ) )
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:59_1_2"><i><font color="Green">E35</font></i></a></i>;<br/>
<a NAME="E4:59_1_2"/><i><font color="Green">E38</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a>  &amp; <font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font></span><a href="domain_1.html#K1">]</a></span> where B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, B is    <a href="subset_1.html#M1">Element</a> of  <a href="finsub_1.html#K5">Fin</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> : ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> ) ) </span>}</span>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_2.html#K3">.</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>] ) )
 
<b>by </b><i><a class="txt" href="bagorder.html#E2:59_1_2"><i><font color="Green">E36</font></i></a></i>;<br/>
<a NAME="E5:59_1_2"/><i><font color="Green">E39</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>, <font color="Olive">b<sub>3</sub></font>, <font color="Olive">b<sub>4</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">S<sub>1</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>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>4</sub></font>] holds <br/><font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>4</sub></font>
 
;<br/>
<b>thus </b><a NAME="E6:59_1_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>6</sub></font>
 <b>from </b><i><a class="ref" href="recdef_1.html#S9">RECDEF_1:sch 9</a>(<a class="txt" href="bagorder.html#E3:59_1_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="bagorder.html#E4:59_1_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="bagorder.html#E5:59_1_2"><i><font color="Green">E39</font></i></a>);</i><br/>


</div>
