<?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="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>



<b>assume </b><a NAME="E1:54"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V4" target="_self">Dickson</a>
 ;<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>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>,  <a href="nat_1.html#NM1">Nat</a>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">a<sub>1</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,  <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#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font> being non <a href="xboole_0.html#V1">empty</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> 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"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) </span>}</span>  &amp; not <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>3</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>3</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Olive">b<sub>2</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>3</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>4</sub></font>,<font color="Maroon">a<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">a<sub>2</sub></font> );<br/>
<b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="hidden.html#M1">set</a> ,  <a href="nat_1.html#NM1">Nat</a>,   <a href="hidden.html#M1">set</a> ] means ( not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a> implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">a<sub>2</sub></font>,<font color="Maroon">a<sub>3</sub></font>] );<br/>
<a NAME="E2:54"/><i><font color="Green">E39</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a><br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ex <font color="Olive">b<sub>3</sub></font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  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>,<font color="Olive">b<sub>3</sub></font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>



<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:54_1_1"/>
<font color="Maroon">c<sub>7</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="dickson.html#E3:54_1_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  as   <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="tarski.html#D3">TARSKI:def 3</a></i>;<br/>
<b>set </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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_1_2"/>
( <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> or  not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_1"><b>suppose </b></a><a NAME="E1:54_1_2_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>take </b>
1
;<br/><b>assume </b><a NAME="E2:54_1_2_1"/>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54_1_2_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>9</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E6:54_1_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 <b>and </b><br/><a NAME="E7:54_1_2_1"/><i><font color="Green">E42</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_1_2_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E8:54_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E7:54_1_2_1"><i><font color="Green">E42</font></i></a></i>;<br/><b>hence </b><a NAME="E9:54_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>5</sub></font>,1]
 <b>by </b><i><a class="txt" href="dickson.html#E1:54_1_2_1"><i><font color="Green">E40</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_1_2_2"><b>suppose </b></a><a NAME="E1:54_1_2_2"/><i><font color="Green">E40</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>10</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:54_1_2_2"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E4:54_1_2_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E5:54_1_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E3:54_1_2_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E6:54_1_2_2"/><b>then </b>
<font color="Maroon">c<sub>11</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_1_2_2"><i><font color="Green">E42</font></i></a></i>;<br/><a NAME="E7:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:54_1_2_2"/><i><font color="Green">E43</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E10:54_1_2_2"/>
<font color="Maroon">c<sub>12</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E11:54_1_2_2"/><i><font color="Green">E44</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E9:54_1_2_2"><i><font color="Green">E43</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E12:54_1_2_2"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span>  as  non <a href="xboole_0.html#V1">empty</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="txt" href="dickson.html#E1:54_1_2_2"><i><font color="Green">E40</font></i></a>, <a class="txt" href="dickson.html#E11:54_1_2_2"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T27" target="_self">Th27</a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>set </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>5</sub></font>;<br/><b>take </b>
<font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>5</sub></font>
;<br/><b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><b>assume </b><a NAME="E14:54_1_2_2"/><i><font color="Green">E45</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>take </b>
<font color="Maroon">c<sub>7</sub></font>
;<br/><b>reconsider </b><font color="Maroon">c<sub>16</sub></font> = <font color="Maroon">c<sub>13</sub></font> as  non <a href="xboole_0.html#V1">empty</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> ;<br/><b>take </b>
<font color="Maroon">c<sub>16</sub></font>
;<br/><b>thus </b><a NAME="E16:54_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><b>thus </b><a NAME="E17:54_1_2_2"/>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>5</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_1_2_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>thus </b><a NAME="E18:54_1_2_2"/>
<font color="Maroon">c<sub>16</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>7</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 ;<br/><b>thus </b><a NAME="E19:54_1_2_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>13</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>16</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>5</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>5</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>
<b>consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E4:54"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>and </b><br/><a NAME="E5:54"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>4</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>4</sub></font> as  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a>  <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E40</font></i></a>, <a class="ref" href="dickson.html#T27" target="_self">Th27</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>8</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E8:54"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K7" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>
 <b>and </b><br/><a NAME="E9:54"/><i><font color="Green">E43</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>2</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</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="dickson.html#E2:54"><i><font color="Green">E39</font></i></a>);<br/></i>
<a NAME="E10:54"/><i><font color="Green">E44</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</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="E11:54"/><i><font color="Green">E45</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>8</sub></font> <a href="tarski.html#R1">c=</a>  <a href="numbers.html#K5">NAT</a> 
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2"><b>now </b></a><div class="add"><a NAME="E1:54_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E5:54"><i><font color="Green">E41</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_2"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> holds <br/><font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> ;<br/><a NAME="E3:54_2"/><i><font color="Green">E48</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_2"><i><font color="Green">E46</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:54_2_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>11</sub></font>)
 <b>and </b><br/><a NAME="E4:54_2_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><b>thus </b><a NAME="E5:54_2_1"/>
<font color="Maroon">c<sub>10</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_2"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E3:54_2_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="dickson.html#E4:54_2_1"><i><font color="Green">E50</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_2"/><b>then </b><i><font color="Green">E49</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_2"><i><font color="Green">E48</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>10</sub></font> as   <a href="nat_1.html#NM1">Nat</a> ;<br/><a NAME="E3:54_2_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</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="E4:54_2_2"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E5:54_2_2"/><b>then </b><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>13</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E8:54_2_2"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E9:54_2_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E40</font></i></a>, <a class="txt" href="dickson.html#E5:54_2_2"><i><font color="Green">E50</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E10:54_2_2"/>
<font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54"><i><font color="Green">E40</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>13</sub></font> as    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E8:54_2_2"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E12:54_2_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>14</sub></font>)
 <b>by </b><i><a class="txt" href="dickson.html#E9:54_2_2"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E13:54_2_2"/>
<font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>14</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E8:54_2_2"><i><font color="Green">E51</font></i></a></i>;<br/><b>hence </b><a NAME="E14:54_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E12:54_2_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E7:54_2"/><b>then </b>
 <a href="numbers.html#K5">NAT</a>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>hence </b><a NAME="E8:54_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:54_2"><i><font color="Green">E49</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>9</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E14:54"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>9</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E15:54"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>9</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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E14:54"><i><font color="Green">E46</font></i></a></i>;<br/>
<a NAME="E16:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E18:54"/><i><font color="Green">E48</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E19:54"/><i><font color="Green">E49</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E21:54"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>and </b><br/><a NAME="E22:54"/>
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>11</sub></font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E19:54"><i><font color="Green">E49</font></i></a></i>;<br/>
<a NAME="E23:54"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17">NORMSP_1:17</a></i>;<br/>
<a NAME="E24:54"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E8:54"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dickson.html#E18:54"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E21:54"><i><font color="Green">E50</font></i></a>, <a class="ref" href="dickson.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E25:54"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <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="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E15:54"><i><font color="Green">E47</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>12</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E27:54"/><i><font color="Green">E52</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>5</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>2</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E28:54"/><i><font color="Green">E53</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font> &amp; not <font color="Maroon">H<sub>2</sub></font>(<font color="Olive">b<sub>1</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<b>consider </b><font color="Maroon">c<sub>13</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E30:54"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>12</sub></font>
 <b>and </b><br/><a NAME="E31:54"/><i><font color="Green">E55</font></i>: 
not <font color="Maroon">H<sub>2</sub></font>(<font color="Maroon">c<sub>13</sub></font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E28:54"><i><font color="Green">E53</font></i></a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>14</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>15</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>16</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> ;<br/>
<a NAME="E32:54"/><i><font color="Green">E56</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>from </b><i><a class="ref" href="finseq_1.html#S6">FINSEQ_1:sch 6</a>();<br/></i>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_3_1"/>
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_3_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>and </b><br/><a NAME="E4:54_3_1"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/><a NAME="E5:54_3_1"/><i><font color="Green">E59</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E6:54_3_1"/>
( <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 or <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 )
 ;<br/><a NAME="E7:54_3_1"/><b>then </b>
( <font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_1"><i><font color="Green">E57</font></i></a>, <a class="txt" href="dickson.html#E4:54_3_1"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E5:54_3_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>hence </b><a NAME="E8:54_3_1"/>
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_3"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_3_2"/>
( <font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="dickson.html#E2:54_3"><i><font color="Green">E57</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_1"><b>suppose </b></a><a NAME="E1:54_3_2_1"/>
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_3_2_1"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_1"/>
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_2_1"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_3_2_2"><b>suppose </b></a><a NAME="E1:54_3_2_2"/>
<font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>18</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_3_2_2"/><i><font color="Green">E58</font></i>: 
( <font color="Maroon">c<sub>17</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>18</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>18</sub></font> &amp; <font color="Maroon">c<sub>18</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>18</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:54_3_2_2"/>
<font color="Maroon">c<sub>17</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="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_3_2_2"><i><font color="Green">E58</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<a NAME="E34:54"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 &amp; <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<a NAME="E35:54"/><b>then </b><i><font color="Green">E57</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E24:54"><i><font color="Green">E51</font></i></a>, <a class="txt" href="dickson.html#E27:54"><i><font color="Green">E52</font></i></a>, <a class="txt" href="dickson.html#E30:54"><i><font color="Green">E54</font></i></a>, <a class="txt" href="dickson.html#E31:54"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E32:54"><i><font color="Green">E56</font></i></a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</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>];<br/>
<a NAME="E36:54"/><i><font color="Green">E58</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54"><i><font color="Green">E43</font></i></a>, <a class="txt" href="dickson.html#E35:54"><i><font color="Green">E57</font></i></a></i>;<br/>
<div><i><font color="Green">E59</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>17</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:54_4"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>17</sub></font>]
 ;<br/><b>consider </b><font color="Maroon">c<sub>18</sub></font> being   <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>, <font color="Maroon">c<sub>19</sub></font> being  non <a href="xboole_0.html#V1">empty</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> such that </b><br/><a NAME="E3:54_4"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E4:54_4"/><i><font color="Green">E62</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E5:54_4"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E6:54_4"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54_4"><i><font color="Green">E60</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>20</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>21</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>22</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>23</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E8:54_4"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>23</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>23</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E9:54_4"/>
<font color="Maroon">c<sub>23</sub></font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E8:54_4"><i><font color="Green">E65</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E10:54_4"/><b>then </b>
<font color="Maroon">c<sub>23</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E8:54_4"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E11:54_4"/><b>then </b>
<font color="Maroon">c<sub>19</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E5:54_4"><i><font color="Green">E63</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>24</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E13:54_4"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>24</sub></font> &amp; <font color="Maroon">c<sub>24</sub></font> is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E14:54_4"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>18</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E13:54_4"><i><font color="Green">E66</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1"><b>now </b></a><div class="add"><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>);<br/><a NAME="E1:54_4_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>19</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E13:54_4"><i><font color="Green">E66</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_4_1"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span> ;<br/><a NAME="E3:54_4_1"/><i><font color="Green">E70</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_4_1"><i><font color="Green">E68</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_4_1_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:54_4_1_1"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>)
 <b>and </b><br/><a NAME="E4:54_4_1_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><b>thus </b><a NAME="E5:54_4_1_1"/>
<font color="Maroon">c<sub>26</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_4_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E3:54_4_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_1_1"><i><font color="Green">E72</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_4_1"/><b>then </b><i><font color="Green">E71</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1"><i><font color="Green">E70</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_4_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>26</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_4_1_2"/>
<font color="Maroon">c<sub>26</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_4_1_2"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E4:54_4_1_2"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E5:54_4_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>27</sub></font>
 ;<br/>
<a NAME="E6:54_4_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_1_2"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E5:54_4_1_2"><i><font color="Green">E74</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>28</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <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/>
<b>consider </b><font color="Maroon">c<sub>29</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:54_4_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 <b>and </b><br/><a NAME="E10:54_4_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green">E67</font></i></a>, <a class="txt" href="dickson.html#E6:54_4_1_2"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E11:54_4_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E12:54_4_1_2"/>
<font color="Maroon">c<sub>18</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_4_1_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_4_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E4:54_4_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E5:54_4_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>31</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E6:54_4_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</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/>
<b>hence </b><a NAME="E7:54_4_1_2_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1_2_1"><i><font color="Green">E79</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_4_1_2"/><b>then </b>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_4_1_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>30</sub></font> = <font color="Maroon">c<sub>29</sub></font> as    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_4_1_2"><i><font color="Green">E76</font></i></a></i>;<br/>
<a NAME="E15:54_4_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_4_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E10:54_4_1_2"><i><font color="Green">E77</font></i></a></i>;<br/>
<a NAME="E16:54_4_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>30</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_4_1_2"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_4_1_2"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_1_2"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E15:54_4_1_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_4"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E5:54_4_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>25</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E17:54_4"/><i><font color="Green">E68</font></i>: 
( <font color="Maroon">c<sub>25</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>25</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E18:54_4"/>
<font color="Maroon">c<sub>25</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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E17:54_4"><i><font color="Green">E68</font></i></a></i>;<br/><a NAME="E19:54_4"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font><b> such that </b><br/><a NAME="E21:54_4"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E22:54_4"/><i><font color="Green">E70</font></i>: 
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><b>consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E24:54_4"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>26</sub></font>
 <b>and </b><br/><a NAME="E25:54_4"/><i><font color="Green">E72</font></i>: 
not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>27</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E22:54_4"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E26:54_4"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>19</sub></font>
 ;<br/><a NAME="E27:54_4"/>
<font color="Maroon">c<sub>19</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_4"><i><font color="Green">E67</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><a NAME="E28:54_4"/><b>then </b>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E26:54_4"><i><font color="Green">E73</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E30:54_4"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E31:54_4"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>28</sub></font>
 <b>and </b><br/><a NAME="E32:54_4"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>28</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4"><i><font color="Green">E61</font></i></a></i>;<br/><a NAME="E33:54_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>19</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E21:54_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E30:54_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E32:54_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self">Def12</a></i>;<br/><a NAME="E34:54_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_4"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E21:54_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E30:54_4"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E32:54_4"><i><font color="Green">E75</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self">Def12</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">a<sub>1</sub></font>;<br/><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>30</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>31</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> ;<br/><a NAME="E35:54_4"/><i><font color="Green">E78</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="dickson.html#S1" target="_self">DICKSON:sch 1</a>();<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_4_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>32</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:54_4_2_1"/>
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_4_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font>
 <b>and </b><br/><a NAME="E4:54_4_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>33</sub></font> )
 ;<br/><a NAME="E5:54_4_2_1"/>
( <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> or <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 ;<br/><a NAME="E6:54_4_2_1"/><b>then </b>
( <font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  )
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_1"><i><font color="Green">E79</font></i></a>, <a class="txt" href="dickson.html#E4:54_4_2_1"><i><font color="Green">E80</font></i></a></i>;<br/><b>hence </b><a NAME="E7:54_4_2_1"/>
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:54_4_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:54_4_2_2"/>
( <font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  or <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span>  )
 </span><b>by </b><i><a class="txt" href="dickson.html#E2:54_4_2"><i><font color="Green">E79</font></i></a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_1"><b>suppose </b></a><a NAME="E1:54_4_2_2_1"/>
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_4_2_2_1"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>33</sub></font> )
 ;<br/><a NAME="E4:54_4_2_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>33</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E34:54_4"><i><font color="Green">E77</font></i></a>, <a class="txt" href="dickson.html#E3:54_4_2_2_1"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E5:54_4_2_2_1"/>
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_2_1"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="54_4_2_2_2"><b>suppose </b></a><a NAME="E1:54_4_2_2_2"/>
<font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_4_2_2_2"/><i><font color="Green">E80</font></i>: 
( <font color="Maroon">c<sub>32</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>33</sub></font> &amp; <font color="Maroon">c<sub>33</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>33</sub></font> )
 ;<br/><b>thus </b><a NAME="E4:54_4_2_2_2"/>
<font color="Maroon">c<sub>32</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_4_2_2_2"><i><font color="Green">E80</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E37:54_4"/><b>then </b>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="xboole_0.html#K2">\/</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <font color="Maroon">c<sub>17</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">b<sub>1</sub></font>] ) </span>}</span> 
 <b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/><a NAME="E38:54_4"/><b>then </b>
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="recdef_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E21:54_4"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E24:54_4"><i><font color="Green">E71</font></i></a>, <a class="txt" href="dickson.html#E25:54_4"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E33:54_4"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E35:54_4"><i><font color="Green">E78</font></i></a>, <a class="ref" href="finset_1.html#T14">FINSET_1:14</a></i>;<br/><b>hence </b><a NAME="E39:54_4"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">c<sub>17</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="txt" href="dickson.html#E9:54"><i><font color="Green">E43</font></i></a></i>;<br/></div><b>end;</b></div>
<a NAME="E38:54"/><i><font color="Green">E60</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>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="dickson.html#E36:54"><i><font color="Green">E58</font></i></a>, <a class="txt" href="dickson.html#E37:54"><i><font color="Green">E59</font></i></a>);<br/></i>
<b>set </b><font color="Maroon">c<sub>17</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>take </b>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font>
;<br/>

<b>reconsider </b><font color="Maroon">c<sub>18</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="numbers.html#K1">REAL</a>  <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5"><b>now </b></a><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1"><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/><a NAME="E1:54_5_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>20</sub></font> = <font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="dickson.html#E11:54"><i><font color="Green">E45</font></i></a></i>;<br/><b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <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>, <font color="Maroon">c<sub>22</sub></font> being  non <a href="xboole_0.html#V1">empty</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> such that </b><br/><a NAME="E4:54_5_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E5:54_5_1"/><i><font color="Green">E62</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E6:54_5_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E7:54_5_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">c<sub>18</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> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<font color="Maroon">c<sub>20</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E38:54"><i><font color="Green">E60</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>24</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>25</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/><b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E9:54_5_1"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E10:54_5_1"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E9:54_5_1"><i><font color="Green">E65</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E11:54_5_1"/><b>then </b>
<font color="Maroon">c<sub>26</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1"><i><font color="Green">E65</font></i></a></i>;<br/><a NAME="E12:54_5_1"/><b>then </b>
<font color="Maroon">c<sub>22</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E6:54_5_1"><i><font color="Green">E63</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E14:54_5_1"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>27</sub></font>
 <b>and </b><br/><a NAME="E15:54_5_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><a NAME="E16:54_5_1"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E14:54_5_1"><i><font color="Green">E66</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/><a NAME="E17:54_5_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1"><b>now </b></a><div class="add"><a NAME="E1:54_5_1_1"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E14:54_5_1"><i><font color="Green">E66</font></i></a>, <a class="txt" href="dickson.html#E15:54_5_1"><i><font color="Green">E67</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_5_1_1"/><i><font color="Green">E71</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font>);<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><a NAME="E3:54_5_1_1"/><i><font color="Green">E72</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_5_1_1"><i><font color="Green">E70</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_5_1_1_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:54_5_1_1_1"/><i><font color="Green">E73</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>20</sub></font>)
 <b>and </b><br/><a NAME="E4:54_5_1_1_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>thus </b><a NAME="E5:54_5_1_1_1"/>
<font color="Maroon">c<sub>30</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_5_1_1"><i><font color="Green">E71</font></i></a>, <a class="txt" href="dickson.html#E3:54_5_1_1_1"><i><font color="Green">E73</font></i></a>, <a class="txt" href="dickson.html#E4:54_5_1_1_1"><i><font color="Green">E74</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_5_1_1"/><b>then </b><i><font color="Green">E73</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_5_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_5_1_1_2"/>
<font color="Maroon">c<sub>30</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_5_1_1_2"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/>
<a NAME="E6:54_5_1_1_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E4:54_5_1_1_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E5:54_5_1_1_2"><i><font color="Green">E76</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:54_5_1_1_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E10:54_5_1_1_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:54_5_1"><i><font color="Green">E68</font></i></a>, <a class="txt" href="dickson.html#E6:54_5_1_1_2"><i><font color="Green">E77</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E11:54_5_1_1_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E16:54_5_1"><i><font color="Green">E68</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E12:54_5_1_1_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_5_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_5_1_1_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_5_1_1_2_1"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E4:54_5_1_1_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E5:54_5_1_1_2_1"/>
<font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green">E61</font></i></a></i>;<br/>
<a NAME="E6:54_5_1_1_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</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/>
<b>hence </b><a NAME="E7:54_5_1_1_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1_2_1"><i><font color="Green">E81</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_5_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_5_1_1_2"><i><font color="Green">E80</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<a NAME="E15:54_5_1_1_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>20</sub></font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_5_1_1_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E10:54_5_1_1_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E16:54_5_1_1_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_5_1_1_2"><i><font color="Green">E78</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_5_1_1_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_5_1_1_2"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E15:54_5_1_1_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_5_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:54_5_1"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E5:54_5_1_1"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div><b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E20:54_5_1"/><i><font color="Green">E70</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E21:54_5_1"/>
<font color="Maroon">c<sub>28</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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E20:54_5_1"><i><font color="Green">E70</font></i></a></i>;<br/><a NAME="E22:54_5_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font><b> such that </b><br/><a NAME="E24:54_5_1"/><i><font color="Green">E71</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font>
 <b>and </b><br/><a NAME="E25:54_5_1"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/><a NAME="E26:54_5_1"/>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E17:54_5_1"><i><font color="Green">E69</font></i></a>, <a class="txt" href="dickson.html#E24:54_5_1"><i><font color="Green">E71</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E28:54_5_1"/><i><font color="Green">E72</font></i>: 
(  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font> &amp; <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>30</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E4:54_5_1"><i><font color="Green">E61</font></i></a></i>;<br/><b>thus </b><a NAME="E29:54_5_1"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>18</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> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:54_5_1"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E28:54_5_1"><i><font color="Green">E72</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self">Def12</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E2:54_5"/>
<font color="Maroon">c<sub>18</sub></font> is <a href="seqm_3.html#V1">increasing</a>
 <b>by </b><i><a class="ref" href="seqm_3.html#D11">SEQM_3:def 11</a></i>;<br/><b>let </b><font color="Maroon">c<sub>19</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E3:54_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>18</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><b>hence </b><a NAME="E4:54_5"/>
<font color="Maroon">c<sub>18</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>19</sub></font> is   <a href="nat_1.html#NM1">Nat</a>
 <b>by </b><i><a class="txt" href="dickson.html#E11:54"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>19</sub></font> = <font color="Maroon">c<sub>18</sub></font> as  <a href="seqm_3.html#V1">increasing</a> <a href="seqm_3.html#NM1">Seq_of_Nat</a> <b>by </b><i><a class="ref" href="seqm_3.html#T29">SEQM_3:29</a></i>;<br/>
<a NAME="E42:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>19</sub></font>
 
;<br/>
<b>hence </b><a NAME="E43:54"/>
<font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font> is    <a href="bhsp_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="bhsp_3.html#D4">BHSP_3:def 4</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>20</sub></font> be   <a href="nat_1.html#NM1">Nat</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="dickson.html#D2" target="_self">DICKSON:def 2</a><br/>

<a NAME="E44:54"/><i><font color="Green">E61</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<a NAME="E45:54"/><i><font color="Green">E62</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E10:54"><i><font color="Green">E44</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>21</sub></font> being   <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>, <font color="Maroon">c<sub>22</sub></font> being  non <a href="xboole_0.html#V1">empty</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> such that </b><br/><a NAME="E47:54"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">c<sub>21</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font></span>)</span> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 <b>and </b><br/><a NAME="E48:54"/><i><font color="Green">E64</font></i>: 
not <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>and </b><br/><a NAME="E49:54"/><i><font color="Green">E65</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 <b>and </b><br/><a NAME="E50:54"/><i><font color="Green">E66</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="dickson.html#K8" target="_self">mindex</a> <span class="p1">(<span class="default"><a href="subset_1.html#K10">choose</a> <span class="p2">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> </span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E38:54"><i><font color="Green">E60</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>23</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>24</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>25</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>26</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E52:54"/><i><font color="Green">E67</font></i>: 
( <font color="Maroon">c<sub>26</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>26</sub></font> is <a href="finset_1.html#V1">finite</a> )
 <b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E53:54"/>
<font color="Maroon">c<sub>26</sub></font> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E52:54"><i><font color="Green">E67</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/>
<a NAME="E54:54"/><b>then </b>
<font color="Maroon">c<sub>26</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E52:54"><i><font color="Green">E67</font></i></a></i>;<br/>
<a NAME="E55:54"/><b>then </b>
<font color="Maroon">c<sub>22</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="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font> : <font color="Olive">b<sub>1</sub></font> is <a href="finset_1.html#V1">finite</a> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E49:54"><i><font color="Green">E65</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>27</sub></font> being    <a href="subset_1.html#M2">Element</a> of  <a href="dickson.html#K9" target="_self">Dickson-bases</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E57:54"/><i><font color="Green">E68</font></i>: 
<font color="Maroon">c<sub>27</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E58:54"/><i><font color="Green">E69</font></i>: 
<font color="Maroon">c<sub>27</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/>
<a NAME="E59:54"/><i><font color="Green">E70</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="dickson.html#R1" target="_self">is_Dickson-basis_of</a> <font color="Maroon">c<sub>21</sub></font>,<font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:54"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E57:54"><i><font color="Green">E68</font></i></a>, <a class="ref" href="dickson.html#D13" target="_self">Def13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6"><b>now </b></a><div class="add"><a NAME="E1:54_6"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">c<sub>22</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E57:54"><i><font color="Green">E68</font></i></a>, <a class="txt" href="dickson.html#E58:54"><i><font color="Green">E69</font></i></a></i>;<br/><b>assume </b><a NAME="E2:54_6"/><i><font color="Green">E72</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> holds <br/><font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a>
 ;<br/><b>deffunc </b><font color="Maroon">H<sub>4</sub></font>(   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>) <b>-&gt; </b>   <a href="hidden.html#M1">set</a>  = <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">a<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>);<br/><b>set </b><font color="Maroon">c<sub>28</sub></font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> ;<br/><b>set </b><font color="Maroon">c<sub>29</sub></font> = <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> ;<br/><a NAME="E3:54_6"/><i><font color="Green">E73</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>from </b><i><a class="ref" href="fraenkel.html#S21">FRAENKEL:sch 21</a>(<a class="txt" href="dickson.html#E1:54_6"><i><font color="Green">E71</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:54_6_1"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 ;<br/><b>then consider </b><font color="Maroon">c<sub>31</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E3:54_6_1"/><i><font color="Green">E74</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>31</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>)
 <b>and </b><br/><a NAME="E4:54_6_1"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 ;<br/><b>thus </b><a NAME="E5:54_6_1"/>
<font color="Maroon">c<sub>30</sub></font> is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E2:54_6"><i><font color="Green">E72</font></i></a>, <a class="txt" href="dickson.html#E3:54_6_1"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E4:54_6_1"><i><font color="Green">E75</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E5:54_6"/><b>then </b><i><font color="Green">E74</font></i>: 
 <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span>  is <a href="finset_1.html#V1">finite</a>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6"><i><font color="Green">E73</font></i></a>, <a class="ref" href="finset_1.html#T25">FINSET_1:25</a></i>;<br/><a NAME="E6:54_6"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>30</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_6_2"/>
<font color="Maroon">c<sub>30</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="nat_1.html#NM1">Nat</a> : ( <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>31</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_6_2"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E4:54_6_2"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font>
 <b>and </b><br/><a NAME="E5:54_6_2"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>31</sub></font>
 ;<br/>
<a NAME="E6:54_6_2"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green">E63</font></i></a>, <a class="txt" href="dickson.html#E4:54_6_2"><i><font color="Green">E76</font></i></a>, <a class="txt" href="dickson.html#E5:54_6_2"><i><font color="Green">E77</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>32</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>31</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/>
<b>consider </b><font color="Maroon">c<sub>33</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E9:54_6_2"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 <b>and </b><br/><a NAME="E10:54_6_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c<sub>33</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>32</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green">E70</font></i></a>, <a class="txt" href="dickson.html#E6:54_6_2"><i><font color="Green">E78</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E11:54_6_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green">E70</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E12:54_6_2"/>
<font color="Maroon">c<sub>21</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="54_6_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>34</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:54_6_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>35</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:54_6_2_1"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E4:54_6_2_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>35</sub></font>
 <b>and </b><br/><a NAME="E5:54_6_2_1"/>
<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>35</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E6:54_6_2_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>2</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/>
<b>hence </b><a NAME="E7:54_6_2_1"/>
<font color="Maroon">c<sub>34</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6_2_1"><i><font color="Green">E82</font></i></a>, <a class="ref" href="funct_1.html#T12">FUNCT_1:12</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E13:54_6_2"/><b>then </b>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E11:54_6_2"><i><font color="Green">E81</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>34</sub></font> = <font color="Maroon">c<sub>33</sub></font> as    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E9:54_6_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<a NAME="E15:54_6_2"/><i><font color="Green">E82</font></i>: 
<font color="Maroon">c<sub>31</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>)
 
<b>by </b><i><a class="txt" href="dickson.html#E5:54_6_2"><i><font color="Green">E77</font></i></a>, <a class="txt" href="dickson.html#E10:54_6_2"><i><font color="Green">E80</font></i></a></i>;<br/>
<a NAME="E16:54_6_2"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>34</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E9:54_6_2"><i><font color="Green">E79</font></i></a></i>;<br/>
<b>hence </b><a NAME="E17:54_6_2"/>
<font color="Maroon">c<sub>30</sub></font> <a href="hidden.html#R2">in</a>  <a href="tarski.html#K3">union</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>4</sub></font>(<font color="Olive">b<sub>1</sub></font>) where B is    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> : <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> </span>}</span> 
 <b>by </b><i><a class="txt" href="dickson.html#E3:54_6_2"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E15:54_6_2"><i><font color="Green">E82</font></i></a>, <a class="ref" href="tarski.html#D4">TARSKI:def 4</a></i>;<br/>


</div><b>end;</b></div><b>hence </b><a NAME="E7:54_6"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E48:54"><i><font color="Green">E64</font></i></a>, <a class="txt" href="dickson.html#E5:54_6"><i><font color="Green">E74</font></i></a>, <a class="ref" href="finset_1.html#T13">FINSET_1:13</a></i>;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>28</sub></font> being    <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font><b> such that </b><br/><a NAME="E62:54"/><i><font color="Green">E71</font></i>: 
( <font color="Maroon">c<sub>28</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>28</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E63:54"/>
<font color="Maroon">c<sub>28</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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="dickson.html#E62:54"><i><font color="Green">E71</font></i></a></i>;<br/>
<a NAME="E64:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <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="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 
;<br/>
<b>then consider </b><font color="Maroon">c<sub>29</sub></font> being    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font><b> such that </b><br/><a NAME="E66:54"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">c<sub>29</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span> 
 <b>and </b><br/><a NAME="E67:54"/>
ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>29</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> )
 ;<br/>
<a NAME="E68:54"/><i><font color="Green">E73</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>22</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E66:54"><i><font color="Green">E72</font></i></a></i>;<br/>
<a NAME="E69:54"/>
<font color="Maroon">c<sub>22</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E59:54"><i><font color="Green">E70</font></i></a>, <a class="ref" href="dickson.html#D9" target="_self">Def9</a></i>;<br/>
<a NAME="E70:54"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>21</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E68:54"><i><font color="Green">E73</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>30</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E72:54"/><i><font color="Green">E74</font></i>: 
 <a href="subset_1.html#K10">choose</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is    <a href="struct_0.html#M1">Element</a> of <font color="Maroon">c<sub>22</sub></font> : ex <font color="Olive">b<sub>1</sub></font> being   <a href="struct_0.html#M1">Element</a> of  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font> &amp; not <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>2</sub></font>,<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font>) is <a href="finset_1.html#V1">finite</a> ) </span>}</span>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>and </b><br/><a NAME="E73:54"/><i><font color="Green">E75</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>30</sub></font>
 <b>and </b><br/><a NAME="E74:54"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">c<sub>19</sub></font> <a href="seqm_3.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>30</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E47:54"><i><font color="Green">E63</font></i></a></i>;<br/>
<a NAME="E75:54"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font> <a href="orders_2.html#R1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E44:54"><i><font color="Green">E61</font></i></a>, <a class="txt" href="dickson.html#E45:54"><i><font color="Green">E62</font></i></a>, <a class="txt" href="dickson.html#E50:54"><i><font color="Green">E66</font></i></a>, <a class="txt" href="dickson.html#E72:54"><i><font color="Green">E74</font></i></a>, <a class="txt" href="dickson.html#E73:54"><i><font color="Green">E75</font></i></a>, <a class="txt" href="dickson.html#E74:54"><i><font color="Green">E76</font></i></a>, <a class="ref" href="dickson.html#D12" target="_self">Def12</a></i>;<br/>
<b>hence </b><a NAME="E76:54"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>20</sub></font></span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="normsp_1.html#K2">.</a> <span class="p3">(<span class="default"><font color="Maroon">c<sub>20</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>


</div>
