<?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> =  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>5</sub></font> =  <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>;<br/>
<a NAME="E1:69_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="bagorder.html#K14" target="_self">FinPoset</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="orders_2.html#G1">RelStr</a>(# <span class="p1">(<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="p1">(<span class="default"><a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> #)
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="bagorder.html#K14" target="_self">FinPoset</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as    <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> ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> holds <br/>(  not  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> or <span class="p1"><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> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> );<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> be    <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>;<br/><b>assume </b><a NAME="E1:69_1_1_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><a NAME="E2:69_1_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="card_2.html#T59">CARD_2:59</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>10</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><a NAME="E3:69_1_1_1"/><b>then </b><i><font color="Green">E43</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:69_1_1_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <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> holds <br/>(  not  <a href="card_1.html#K1">Card</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> or <span class="p1"><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> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/><b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be    <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>;<br/><b>assume </b><a NAME="E2:69_1_1_1_2"/><i><font color="Green">E45</font></i>: 
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_1_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_1_1_2_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_1_2_1_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_1_1_2_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1"/>
(  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font> or  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_1"/><i><font color="Green">E48</font></i>: 
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1"/>
(  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font> or  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> )
 </span><b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/><b>then </b>
<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="Maroon">c<sub>11</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</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>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font> <a href="orders_2.html#R3">&lt;=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font>
 ;<br/><div class="add"><a NAME="E2:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/><b>then </b>
<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="Maroon">c<sub>12</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</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>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><b>hence </b><a NAME="E3:69_1_1_1_2_1_2_1_1_2_1_1_1_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2_1_1_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:69_1_1_1_2_1_2_1_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/><a NAME="E2:69_1_1_1_2_1_2_1_1_2_1_1_2"/><i><font color="Green">E49</font></i>: 
 <a href="card_1.html#K1">Card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><span class="p3">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:69_1_1_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E55"><i><font color="Green">Lemma39</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>11</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>, <font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>12</sub></font> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> as    <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>by </b><i><a class="ref" href="bagorder.html#T38" target="_self">Th38</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_2_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 </span><b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="bagorder.html#E2:69_1_1_1_2_1_2_1_1_2_1_1_2"><i><font color="Green">E49</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>15</sub></font>,<font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_1"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>15</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:69_1_1_1_2_1_2_1_1_2_1_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E1:69_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="bagorder.html#E1:69_1_1_1_2_1_2_1_1_2_1_1_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:69_1_1_1_2_1_2_1_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:69_1_1_1_2_1_2_1_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:69_1_1_1_2_1_2"/>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>12</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>11</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E5:69_1_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/><a NAME="E6:69_1_1_1"/><i><font color="Green">E45</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="bagorder.html#E3:69_1_1_1"><i><font color="Green">E43</font></i></a>, <a class="txt" href="bagorder.html#E5:69_1_1_1"><i><font color="Green">E44</font></i></a>);<br/></i><a NAME="E7:69_1_1_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <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>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E9:69_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>10</sub></font> <a href="tarski.html#R2">are_equipotent</a> 
 <b>by </b><i><a class="ref" href="card_1.html#T74">CARD_1:74</a></i>;<br/><a NAME="E10:69_1_1_1"/>
 <a href="card_1.html#K1">Card</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E9:69_1_1_1"><i><font color="Green">E46</font></i></a>, <a class="ref" href="card_1.html#D5">CARD_1:def 5</a></i>;<br/><a NAME="E11:69_1_1_1"/><b>then </b>
( <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>8</sub></font>,<font color="Maroon">c<sub>9</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> or <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>9</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E6:69_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E12:69_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> or <font color="Maroon">c<sub>7</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>6</sub></font> )
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E3:69_1_1"/>
 <a href="bagorder.html#K14" target="_self">FinPoset</a> <font color="Maroon">c<sub>1</sub></font> is <a href="waybel_0.html#V16">connected</a>
 <b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>


</div>
