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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="struct_0.html#V3">empty</a> <a href="waybel_0.html#V16">connected</a> <a href="orders_2.html#NM1">Poset</a>;<br/>

<b>assume </b><a NAME="E1:73"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="wellfnd1.html#V1">well_founded</a>
 ;<br/>

<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/>
<b>assume </b><a NAME="E2:73"/>
not  <a href="bagorder.html#K14" target="_self">FinPoset</a> <font color="Maroon">c<sub>1</sub></font> is <a href="wellfnd1.html#V1">well_founded</a>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E4:73"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>6</sub></font> is <a href="wellfnd1.html#V3">descending</a>
 <b>by </b><i><a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="normsp_1.html#NM2">sequence</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> : <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V3">descending</a> </span>}</span> ;<br/>
<a NAME="E5:73"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="normsp_1.html#NM2">sequence</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> : <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V3">descending</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="normsp_1.html#NM2">sequence</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> : <font color="Olive">b<sub>1</sub></font> is <a href="wellfnd1.html#V3">descending</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  ;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></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 ex <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>3</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>4</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>5</sub></font>, <font color="Olive">b<sub>6</sub></font> being  <a href="normsp_1.html#NM2">sequence</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="mcart_1.html#K2">`2</a>  &amp; ( for <font color="Olive">b<sub>7</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>8</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> st <br/>( <font color="Olive">b<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>7</sub></font> &amp; <font color="Olive">b<sub>8</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="funct_2.html#K3">.</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>8</sub></font> ) ) &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Olive">b<sub>2</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font> <a href="dickson.html#K7">mindex</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>4</sub></font> &amp; ( for <font color="Olive">b<sub>7</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Olive">b<sub>6</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">b<sub>5</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>7</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( for <font color="Olive">b<sub>7</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Olive">b<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>7</sub></font> holds <br/><font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> ) &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<font color="Olive">b<sub>6</sub></font></span><a href="domain_1.html#K1">]</a></span> );<br/>
<a NAME="E7:73"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> 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>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1"><b>proof </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>let </b><font color="Maroon">c<sub>11</sub></font> be    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>;<br/>



<b>set </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="domain_1.html#K3">`2</a> ;<br/>
<a NAME="E1:73_1"/>
<font color="Maroon">c<sub>11</sub></font> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E3:73_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>13</sub></font> is <a href="wellfnd1.html#V3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> <a href="domain_1.html#K3">`2</a>  as   <a href="normsp_1.html#NM2">sequence</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> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green">E48</font></i></a></i>;<br/>
<div><i><font color="Green">E49</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</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> ;<br/><b>assume </b><a NAME="E2:73_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E3:73_1_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font></span>)</span></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#E3:73_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>thus </b><a NAME="E4:73_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="bagorder.html#T37" target="_self">Th37</a></i>;<br/></div><b>end;</b></div>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <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="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<a NAME="E6:73_1"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>15</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</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>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>17</sub></font>;<br/>
<b>take </b>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>17</sub></font>
;<br/>

<b>thus </b><a NAME="E2:73_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font>
 ;<br/>

<b>thus </b><a NAME="E3:73_1_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1"><i><font color="Green">E49</font></i></a></i>;<br/>

<b>thus </b><a NAME="E4:73_1_2"/>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>17</sub></font>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:73_1"/><i><font color="Green">E51</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E6:73_1"><i><font color="Green">E50</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>16</sub></font> =  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>;<br/>
<b>set </b><font color="Maroon">c<sub>18</sub></font> =  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ex <font color="Olive">b<sub>1</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> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> );<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of 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="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><a NAME="E2:73_1_3"/>
<font color="Maroon">c<sub>21</sub></font> <a href="finsub_1.html#K2">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <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>
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</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> ;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>24</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/><b>take </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>22</sub></font>;<br/><b>thus </b><a NAME="E4:73_1_3"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/><b>thus </b><a NAME="E5:73_1_3"/>
<font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>25</sub></font>
 ;<br/></div><b>end;</b></div>
<a NAME="E10:73_1"/><b>then </b><i><font color="Green">E52</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 
;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>;<br/>
<a NAME="E11:73_1"/><i><font color="Green">E53</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E12:73_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> <a href="tarski.html#R1">c=</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="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E13:73_1"/><b>then </b><i><font color="Green">E54</font></i>: 
(  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font> &amp;  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> )
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green">E45</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self">Def19</a></i>;<br/>
<a NAME="E14:73_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="bagorder.html#E11:73_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="dickson.html#D11">DICKSON:def 11</a></i>;<br/>
<div><i><font color="Green">E56</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:73_1_4"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>19</sub></font>]
 ;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1;<br/><b>consider </b><font color="Maroon">c<sub>23</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><b> such that </b><br/><a NAME="E3:73_1_4"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</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><b> such that </b><br/><a NAME="E5:73_1_4"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> &amp; <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>24</sub></font> as   <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/><a NAME="E7:73_1_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</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">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>19</sub></font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E8:73_1_4"/><b>then </b>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</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#E3:73_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green">E58</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green">E59</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font>, <font color="Maroon">c<sub>28</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><b> such that </b><br/><a NAME="E10:73_1_4"/><i><font color="Green">E60</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>and </b><br/><a NAME="E11:73_1_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</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>28</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="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</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> ) or ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</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="Maroon">c<sub>27</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</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="Maroon">c<sub>28</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>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_4_1"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</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>28</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="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</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> ) or ( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</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="Maroon">c<sub>27</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</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="Maroon">c<sub>28</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>  <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#E11:73_1_4"><i><font color="Green">E61</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_1"><b>suppose </b></a><a NAME="E1:73_1_4_1_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_4_1_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_2"><b>suppose </b></a><a NAME="E1:73_1_4_1_2"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</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>28</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="Maroon">c<sub>27</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</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> )
 ;<br/><div class="add"><a NAME="E2:73_1_4_1_2"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E11:73_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E3:73_1_4_1_2"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E13:73_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_4"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green">E58</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green">E60</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_4_1_2"><i><font color="Green">E62</font></i></a>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_4_1_3"><b>suppose </b></a><a NAME="E1:73_1_4_1_3"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</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="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>27</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="Maroon">c<sub>27</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>28</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="Maroon">c<sub>28</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>  <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="E2:73_1_4_1_3"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73_1_4"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_4"><i><font color="Green">E58</font></i></a>, <a class="txt" href="bagorder.html#E5:73_1_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E10:73_1_4"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E16:73_1"/><i><font color="Green">E57</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>4</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#E14:73_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="bagorder.html#E15:73_1"><i><font color="Green">E56</font></i></a>);<br/></i>
<div><i><font color="Green">E58</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:73_1_5"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:73_1_5"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73_1_5"><i><font color="Green">E59</font></i></a>, <a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/><b>thus </b><a NAME="E4:73_1_5"/>
<font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E16:73_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_5"><i><font color="Green">E60</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:73_1_6"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>consider </b><font color="Maroon">c<sub>20</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><b> such that </b><br/><a NAME="E3:73_1_6"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>20</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>take </b><font color="Maroon">c<sub>21</sub></font> = <font color="Maroon">c<sub>20</sub></font>;<br/><b>thus </b><a NAME="E4:73_1_6"/>
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E3:73_1_6"><i><font color="Green">E61</font></i></a></i>;<br/><b>thus </b><a NAME="E5:73_1_6"/>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E17:73_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="bagorder.html#E1:73_1_6"><i><font color="Green">E60</font></i></a>, <a class="txt" href="bagorder.html#E3:73_1_6"><i><font color="Green">E61</font></i></a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green">E60</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_7"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</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>take </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font>;<br/><b>thus </b><a NAME="E2:73_1_7"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/><a NAME="E3:73_1_7"/>
<font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</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><b> such that </b><br/><a NAME="E5:73_1_7"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E18:73_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>thus </b><a NAME="E6:73_1_7"/>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1_7"><i><font color="Green">E61</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self">Def20</a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>19</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</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><b> such that </b><br/><a NAME="E21:73_1"/><i><font color="Green">E61</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>3</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>(<a class="txt" href="bagorder.html#E10:73_1"><i><font color="Green">E52</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>19</sub></font> as   <a href="normsp_1.html#NM2">sequence</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>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E23:73_1"/><i><font color="Green">E62</font></i>: 
 <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> is <a href="wellfnd1.html#V3">descending</a>
 
<b>by </b><i><a class="txt" href="bagorder.html#E3:73_1"><i><font color="Green">E48</font></i></a>, <a class="ref" href="bagorder.html#T41" target="_self">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>22</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>23</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><b> such that </b><br/><a NAME="E2:73_1_8"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>23</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>24</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><b> such that </b><br/><a NAME="E4:73_1_8"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>, <font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <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>27</sub></font> = <font color="Maroon">c<sub>25</sub></font>, <font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>26</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/><a NAME="E7:73_1_8"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>25</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>25</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#E23:73_1"><i><font color="Green">E62</font></i></a>, <a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>29</sub></font>, <font color="Maroon">c<sub>30</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><b> such that </b><br/><a NAME="E9:73_1_8"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> )
 <b>and </b><br/><a NAME="E10:73_1_8"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</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>30</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="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</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> ) or ( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>30</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="Maroon">c<sub>30</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>  <a href="bagorder.html#K13" target="_self">FinOrd</a> <font color="Maroon">c<sub>1</sub></font> ) )
 <b>by </b><i><a class="ref" href="bagorder.html#T40" target="_self">Th40</a></i>;<br/><div><i><font color="Green">E68</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>31</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:73_1_8_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>31</sub></font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="bagorder.html#D20" target="_self">Def20</a></i>;<br/><b>hence </b><a NAME="E2:73_1_8_1"/>
<span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_1"><i><font color="Green">E49</font></i></a></i>;<br/></div><b>end;</b></div><div><i><font color="Green">E69</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_8_2"><b>now </b></a><div class="add"><b>consider </b><font color="Maroon">c<sub>31</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><b> such that </b><br/><a NAME="E2:73_1_8_2"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:73_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>32</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><b> such that </b><br/><a NAME="E4:73_1_8_2"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p3">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p4">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E19:73_1"><i><font color="Green">E60</font></i></a></i>;<br/><b>thus </b><a NAME="E5:73_1_8_2"/>
(  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp;  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_8_2"><i><font color="Green">E70</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8_2"><i><font color="Green">E71</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E13:73_1_8"/>
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  )
 <b>by </b><i><a class="txt" href="bagorder.html#E11:73_1_8"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E14:73_1_8"/><b>then </b>
(  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> &amp;  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green">E69</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self">Def15</a></i>;<br/><b>hence </b><a NAME="E15:73_1_8"/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_8"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8"><i><font color="Green">E64</font></i></a>, <a class="txt" href="bagorder.html#E7:73_1_8"><i><font color="Green">E65</font></i></a>, <a class="ref" href="bagorder.html#T1" target="_self">Th1</a></i>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:73_1_8_3"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a>  or ( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</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>30</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="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</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> ) or ( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>30</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="Maroon">c<sub>30</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>  <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#E10:73_1_8"><i><font color="Green">E67</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_1"><b>suppose </b></a><a NAME="E1:73_1_8_3_1"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_8_3_1"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></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#E9:73_1_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E11:73_1_8"><i><font color="Green">E68</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_2"><b>suppose </b></a><a NAME="E1:73_1_8_3_2"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</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>30</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="Maroon">c<sub>29</sub></font></span>)</span>,<span class="p2">(<span class="default"><a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</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> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:73_1_8_3_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></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#E9:73_1_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="73_1_8_3_3"><b>suppose </b></a><a NAME="E1:73_1_8_3_3"/>
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</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="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>30</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>29</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="Maroon">c<sub>29</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>30</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="Maroon">c<sub>30</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>  <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="E2:73_1_8_3_3"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font></span>)</span></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#E2:73_1_8"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E4:73_1_8"><i><font color="Green">E64</font></i></a>, <a class="txt" href="bagorder.html#E9:73_1_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E12:73_1_8"><i><font color="Green">E69</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E25:73_1"/><b>then </b>
<font color="Maroon">c<sub>20</sub></font> is <a href="wellfnd1.html#V3">descending</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/>
<a NAME="E26:73_1"/><b>then </b>
(  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> )
 
;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>22</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>14</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>15</sub></font>
;<br/>

<b>take </b>
 <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
;<br/>

<b>take </b>
 <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
;<br/>

<b>take </b>
<font color="Maroon">c<sub>20</sub></font>
;<br/>

<b>thus </b><a NAME="E28:73_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>11</sub></font> <a href="domain_1.html#K3">`2</a> 
 ;<br/>

<b>thus </b><a NAME="E29:73_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &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; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</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> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_1"><i><font color="Green">E51</font></i></a></i>;<br/>

<b>thus </b><a NAME="E30:73_1"/>
 <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<b>thus </b><a NAME="E31:73_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/>

<b>thus </b><a NAME="E32:73_1"/>
 <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>consider </b><font color="Maroon">c<sub>24</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><b> such that </b><br/><a NAME="E2:73_1_9"/><i><font color="Green">E63</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E21:73_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>thus </b><a NAME="E3:73_1_9"/>
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E2:73_1_9"><i><font color="Green">E63</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E34:73_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p4">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p5">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <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#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 ;<br/>

<b>thus </b><a NAME="E35:73_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <span class="p1">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p2">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E17:73_1"><i><font color="Green">E58</font></i></a></i>;<br/>

<b>thus </b><a NAME="E36:73_1"/>
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p3">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>20</sub></font></span><a href="domain_1.html#K1">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>11</sub></font> = <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>6</sub></font></span><a href="domain_1.html#K1">]</a></span>;<br/>
<a NAME="E9:73"/>
<font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73"><i><font color="Green">E46</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <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>6</sub></font></span><a href="domain_1.html#K1">]</a></span> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> <b>by </b><i><a class="ref" href="zfmisc_1.html#D2">ZFMISC_1:def 2</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span>, <font color="Maroon">c<sub>14</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>15</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>16</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>17</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>18</sub></font>, <font color="Maroon">c<sub>19</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E12:73"/>
<font color="Maroon">c<sub>14</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> <a href="domain_1.html#K3">`2</a> 
 <b>and </b><br/><a NAME="E13:73"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &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; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</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> )
 <b>and </b><br/><a NAME="E14:73"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>15</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E15:73"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>15</sub></font> <a href="dickson.html#K7">mindex</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E16:73"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>14</sub></font>,<font color="Maroon">c<sub>17</sub></font>
 <b>and </b><br/><a NAME="E17:73"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K6">}</a></span>
 <b>and </b><br/><a NAME="E18:73"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 <b>and </b><br/><a NAME="E19:73"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <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>19</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E7:73"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>20</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,<span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span><b> such that </b><br/><a NAME="E21:73"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 <b>and </b><br/><a NAME="E22:73"/><i><font color="Green">E54</font></i>: 
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>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</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>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>(<a class="txt" href="bagorder.html#E7:73"><i><font color="Green">E47</font></i></a>);<br/></i>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="mcart_1.html#K1">`1</a> ;<br/>
<div><i><font color="Green">E55</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="73_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>21</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:73_2"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="bagorder.html#E1:73_2"><i><font color="Green">E56</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>23</sub></font> = <font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>22</sub></font> as    <a href="subset_1.html#M1">Element</a> of <span class="p1"><a href="zfmisc_1.html#K2">[:</a><span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font></span><a href="zfmisc_1.html#K2">:]</a></span> ;<br/><a NAME="E4:73_2"/>
<font color="Maroon">c<sub>23</sub></font> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>hence </b><a NAME="E5:73_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>21</sub></font>) <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E25:73"/><i><font color="Green">E56</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>21</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="bagorder.html#E23:73"><i><font color="Green">E55</font></i></a>);<br/></i>
<b>reconsider </b><font color="Maroon">c<sub>22</sub></font> = <font color="Maroon">c<sub>21</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being  <a href="normsp_1.html#NM2">sequence</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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; ( for <font color="Olive">b<sub>3</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>2</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>3</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="tarski.html#K4">]</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> ) ) );<br/>
<a NAME="E27:73"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> ;<br/>
<a NAME="E1:73_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E3:73_3"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>25</sub></font> is <a href="wellfnd1.html#V3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  as   <a href="normsp_1.html#NM2">sequence</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> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_3"><i><font color="Green">E58</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>thus </b><a NAME="E5:73_3"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>27</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E6:73_3"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font>
 ;<br/>

<a NAME="E7:73_3"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E19:73"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bagorder.html#E21:73"><i><font color="Green">E53</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E8:73_3"/>
<font color="Maroon">c<sub>19</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>26</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E19:73"><i><font color="Green">E52</font></i></a>, <a class="txt" href="bagorder.html#E21:73"><i><font color="Green">E53</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E9:73_3"/><b>then </b>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E17:73"><i><font color="Green">E50</font></i></a>, <a class="txt" href="bagorder.html#E6:73_3"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E10:73_3"/><b>then </b><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; not <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>16</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E11:73_3"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E7:73_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_3_1"><b>now </b></a><div class="add"><a NAME="E1:73_3_1"/>
( <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:73_3_1"/>
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</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="txt" href="bagorder.html#E6:73_3"><i><font color="Green">E59</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>27</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E14:73_3"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:73"><i><font color="Green">E49</font></i></a>, <a class="txt" href="bagorder.html#E10:73_3"><i><font color="Green">E61</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self">Def20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>29</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><b> such that </b><br/><a NAME="E16:73_3"/><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>14</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>15</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>29</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E13:73"><i><font color="Green">E48</font></i></a></i>;<br/>
<a NAME="E17:73_3"/>
<font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>17</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E18:73_3"/><b>then </b>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E18:73"><i><font color="Green">E51</font></i></a>, <a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green">E64</font></i></a></i>;<br/>
<a NAME="E19:73_3"/><b>then </b>
<font color="Maroon">c<sub>16</sub></font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">c<sub>29</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E20:73_3"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>16</sub></font>,<font color="Maroon">c<sub>27</sub></font></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E7:73_3"><i><font color="Green">E60</font></i></a>, <a class="txt" href="bagorder.html#E11:73_3"><i><font color="Green">E62</font></i></a>, <a class="txt" href="bagorder.html#E14:73_3"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E16:73_3"><i><font color="Green">E64</font></i></a>, <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>
<a NAME="E21:73_3"/><b>then </b>
not <font color="Maroon">c<sub>16</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>28</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E22:73_3"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>16</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>
<b>hence </b><a NAME="E23:73_3"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>27</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E7:73_3"><i><font color="Green">E60</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E28:73"/><i><font color="Green">E58</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:73_4"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">c<sub>23</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>24</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>set </b><font color="Maroon">c<sub>25</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>set </b><font color="Maroon">c<sub>27</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>29</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>30</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>32</sub></font>, <font color="Maroon">c<sub>33</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E4:73_4"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &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; <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</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; <font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>29</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="dickson.html#K7">mindex</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>31</sub></font> &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> ) )
 <b>and </b><br/><a NAME="E5:73_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>33</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E6:73_4"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>34</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E8:73_4"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>34</sub></font> is <a href="wellfnd1.html#V3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>35</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  as   <a href="normsp_1.html#NM2">sequence</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> <b>by </b><i><a class="txt" href="bagorder.html#E8:73_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>35</sub></font>
;<br/>

<b>thus </b><a NAME="E10:73_4"/>
<font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a> 
 ;<br/>

<b>let </b><font color="Maroon">c<sub>36</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E11:73_4"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font>
 ;<br/>

<a NAME="E12:73_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>26</sub></font></span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:73_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<a NAME="E13:73_4"/>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>35</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E5:73_4"><i><font color="Green">E60</font></i></a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>;<br/>
<a NAME="E14:73_4"/><b>then </b>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E11:73_4"><i><font color="Green">E62</font></i></a></i>;<br/>
<a NAME="E15:73_4"/><b>then </b><i><font color="Green">E64</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>32</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> &amp; not <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>30</sub></font></span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<b>hence </b><a NAME="E16:73_4"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
 <b>by </b><i><a class="txt" href="bagorder.html#E12:73_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_4_1"><b>now </b></a><div class="add"><a NAME="E1:73_4_1"/>
( <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>35</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="ref" href="finsub_1.html#D5">FINSUB_1:def 5</a></i>;<br/><b>hence </b><a NAME="E2:73_4_1"/>
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</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="txt" href="bagorder.html#E11:73_4"><i><font color="Green">E62</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>36</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<a NAME="E19:73_4"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E15:73_4"><i><font color="Green">E64</font></i></a>, <a class="ref" href="bagorder.html#D20" target="_self">Def20</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>38</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><b> such that </b><br/><a NAME="E21:73_4"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> &amp; <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>38</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green">E59</font></i></a></i>;<br/>
<a NAME="E22:73_4"/>
<font color="Maroon">c<sub>31</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>24</sub></font> <a href="nat_1.html#K1">+</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T29">NAT_1:29</a></i>;<br/>
<a NAME="E23:73_4"/><b>then </b>
 <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>38</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E4:73_4"><i><font color="Green">E59</font></i></a>, <a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E24:73_4"/><b>then </b>
<font color="Maroon">c<sub>30</sub></font> <a href="waybel_4.html#R2">is_maximal_wrt</a> <font color="Maroon">c<sub>38</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E25:73_4"/><b>then </b>
not <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>36</sub></font></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E12:73_4"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E16:73_4"><i><font color="Green">E65</font></i></a>, <a class="txt" href="bagorder.html#E19:73_4"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E21:73_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="waybel_4.html#D24">WAYBEL_4:def 24</a></i>;<br/>
<a NAME="E26:73_4"/><b>then </b>
not <font color="Maroon">c<sub>30</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<a NAME="E27:73_4"/><b>then </b>
<font color="Maroon">c<sub>37</sub></font> <a href="orders_2.html#R3">&lt;=</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="ref" href="waybel_0.html#D29">WAYBEL_0:def 29</a></i>;<br/>
<b>hence </b><a NAME="E28:73_4"/>
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">c<sub>36</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E12:73_4"><i><font color="Green">E63</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E29:73"/><i><font color="Green">E59</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>2</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#E27:73"><i><font color="Green">E57</font></i></a>, <a class="txt" href="bagorder.html#E28:73"><i><font color="Green">E58</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ex <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</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>ex <font color="Olive">b<sub>2</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>2</sub></font> );<br/>
<a NAME="E30:73"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_5"><b>proof </b></a><div class="add">

<b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> ;<br/>
<a NAME="E1:73_5"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E3:73_5"/><i><font color="Green">E61</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>24</sub></font> is <a href="wellfnd1.html#V3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  as   <a href="normsp_1.html#NM2">sequence</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> <b>by </b><i><a class="txt" href="bagorder.html#E3:73_5"><i><font color="Green">E61</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>25</sub></font>
;<br/>

<a NAME="E5:73_5"/><i><font color="Green">E62</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>27</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>28</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>29</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>30</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E7:73_5"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>and </b><br/><a NAME="E8:73_5"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &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; <font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</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> )
 <b>and </b><br/><a NAME="E9:73_5"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E10:73_5"/>
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font> <a href="dickson.html#K7">mindex</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E11:73_5"/>
<font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>26</sub></font>,<font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E12:73_5"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>31</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>28</sub></font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>29</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>27</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> ) )
 <b>and </b><br/><a NAME="E13:73_5"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>31</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E14:73_5"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font> <a href="tarski.html#R1">c=</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="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E15:73_5"/><b>then </b>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>27</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E9:73_5"><i><font color="Green">E65</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>32</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E17:73_5"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>27</sub></font> &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>28</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>33</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><b> such that </b><br/><a NAME="E19:73_5"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>27</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>33</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E8:73_5"><i><font color="Green">E64</font></i></a>, <a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>32</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>34</sub></font>
;<br/>

<b>thus </b><a NAME="E21:73_5"/>
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> 
 ;<br/>

<a NAME="E22:73_5"/>
<font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E7:73_5"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E17:73_5"><i><font color="Green">E67</font></i></a>, <a class="txt" href="bagorder.html#E19:73_5"><i><font color="Green">E68</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E23:73_5"/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>34</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E5:73_5"><i><font color="Green">E62</font></i></a>, <a class="txt" href="bagorder.html#E13:73_5"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E31:73"/><i><font color="Green">E61</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>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:73_6"/>
<font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">c<sub>23</sub></font>]
 ;<br/>

<b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a> ;<br/>
<a NAME="E2:73_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font>
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E4:73_6"/><i><font color="Green">E62</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>25</sub></font> is <a href="wellfnd1.html#V3">descending</a> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>26</sub></font> = <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  as   <a href="normsp_1.html#NM2">sequence</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> <b>by </b><i><a class="txt" href="bagorder.html#E4:73_6"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>26</sub></font>
;<br/>

<b>set </b><font color="Maroon">c<sub>27</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1 as   <a href="nat_1.html#NM1">Nat</a> ;<br/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>30</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>31</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>32</sub></font> being   <a href="nat_1.html#NM1">Nat</a>, <font color="Maroon">c<sub>33</sub></font>, <font color="Maroon">c<sub>34</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E8:73_6"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>28</sub></font></span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>and </b><br/><a NAME="E9:73_6"/><i><font color="Green">E64</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <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> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> &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; <font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</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> )
 <b>and </b><br/><a NAME="E10:73_6"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K15" target="_self">MinElement</a> <span class="p1">(<span class="default"><a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>30</sub></font></span>)</span>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E11:73_6"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>30</sub></font> <a href="dickson.html#K7">mindex</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E12:73_6"/>
<font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#R2">=</a>  <a href="bagorder.html#K16" target="_self">SeqShift</a> <font color="Maroon">c<sub>29</sub></font>,<font color="Maroon">c<sub>32</sub></font>
 <b>and </b><br/><a NAME="E13:73_6"/>
( ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">c<sub>34</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>33</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> <a href="subset_1.html#K6">\</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">c<sub>31</sub></font></span><a href="domain_1.html#K6">}</a></span> ) &amp; ( for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>32</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>30</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> ) )
 <b>and </b><br/><a NAME="E14:73_6"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>34</sub></font></span><a href="domain_1.html#K1">]</a></span>
 <b>by </b><i><a class="txt" href="bagorder.html#E22:73"><i><font color="Green">E54</font></i></a></i>;<br/>
<a NAME="E15:73_6"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>30</sub></font> <a href="tarski.html#R1">c=</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="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E16:73_6"/><b>then </b>
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">c<sub>30</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green">E45</font></i></a>, <a class="txt" href="bagorder.html#E10:73_6"><i><font color="Green">E65</font></i></a>, <a class="ref" href="bagorder.html#D19" target="_self">Def19</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E18:73_6"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>36</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><b> such that </b><br/><a NAME="E20:73_6"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> &amp; <font color="Maroon">c<sub>36</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>30</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>35</sub></font> <a href="hidden.html#R1">=</a>  <a href="bagorder.html#K11" target="_self">PosetMax</a> <font color="Maroon">c<sub>36</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E9:73_6"><i><font color="Green">E64</font></i></a>, <a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>37</sub></font> = <font color="Maroon">c<sub>35</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green">E67</font></i></a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>37</sub></font>
;<br/>

<b>thus </b><a NAME="E22:73_6"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a> 
 ;<br/>

<a NAME="E23:73_6"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>37</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E8:73_6"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E18:73_6"><i><font color="Green">E67</font></i></a>, <a class="txt" href="bagorder.html#E20:73_6"><i><font color="Green">E68</font></i></a>, <a class="ref" href="bagorder.html#D15" target="_self">Def15</a></i>;<br/>
<a NAME="E24:73_6"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>28</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 
<b>by </b><i><a class="txt" href="bagorder.html#E14:73_6"><i><font color="Green">E66</font></i></a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>;<br/>
<b>hence </b><a NAME="E25:73_6"/>
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>37</sub></font>
 <b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green">E56</font></i></a>, <a class="txt" href="bagorder.html#E23:73_6"><i><font color="Green">E69</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E32:73"/><i><font color="Green">E62</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>3</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#E30:73"><i><font color="Green">E60</font></i></a>, <a class="txt" href="bagorder.html#E31:73"><i><font color="Green">E61</font></i></a>);<br/></i>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">a<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</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> );<br/>
<a NAME="E33:73"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_7"><b>proof </b></a><div class="add">

<a NAME="E1:73_7"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>23</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>24</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:73_7"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>23</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E32:73"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>25</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E5:73_7"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K3">`2</a> 
 <b>and </b><br/><a NAME="E6:73_7"/><i><font color="Green">E67</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>25</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>24</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> 0</span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E29:73"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:73_7"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> 0 &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default">0 <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> 0</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="txt" href="bagorder.html#E1:73_7"><i><font color="Green">E64</font></i></a>, <a class="txt" href="bagorder.html#E3:73_7"><i><font color="Green">E65</font></i></a>, <a class="txt" href="bagorder.html#E5:73_7"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E6:73_7"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E34:73"/><i><font color="Green">E64</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>4</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>4</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="73_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>23</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:73_8"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</sub></font> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>23</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> )
 ;<br/>

<a NAME="E2:73_8"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> 
 
<b>by </b><i><a class="txt" href="bagorder.html#E25:73"><i><font color="Green">E56</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>24</sub></font> being   <a href="normsp_1.html#NM2">sequence</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>, <font color="Maroon">c<sub>25</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E4:73_8"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>24</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>24</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> )
 <b>by </b><i><a class="txt" href="bagorder.html#E32:73"><i><font color="Green">E62</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being   <a href="normsp_1.html#NM2">sequence</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><b> such that </b><br/><a NAME="E6:73_8"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K3">`2</a>  &amp; ( for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>26</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>25</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">b<sub>1</sub></font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="funct_2.html#K3">.</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="domain_1.html#K2">`1</a> </span>)</span></span><a href="tarski.html#K4">]</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="txt" href="bagorder.html#E29:73"><i><font color="Green">E59</font></i></a></i>;<br/>
<b>thus </b><a NAME="E7:73_8"/>
( <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>22</sub></font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>23</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></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="txt" href="bagorder.html#E2:73_8"><i><font color="Green">E65</font></i></a>, <a class="txt" href="bagorder.html#E4:73_8"><i><font color="Green">E66</font></i></a>, <a class="txt" href="bagorder.html#E6:73_8"><i><font color="Green">E67</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E35:73"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>4</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#E33:73"><i><font color="Green">E63</font></i></a>, <a class="txt" href="bagorder.html#E34:73"><i><font color="Green">E64</font></i></a>);<br/></i>
<a NAME="E36:73"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> is <a href="wellfnd1.html#V3">descending</a>
 
<b>by </b><i><a class="ref" href="wellfnd1.html#D7">WELLFND1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E37:73"/>
contradiction
 <b>by </b><i><a class="txt" href="bagorder.html#E1:73"><i><font color="Green">E45</font></i></a>, <a class="ref" href="wellfnd1.html#T15">WELLFND1:15</a></i>;<br/>


</div>
