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<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>assume </b><b>that </b><br/><a NAME="E1:49"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="dickson.html#V3" target="_self">quasi_ordered</a>
 <b>and </b><br/><a NAME="E2:49"/><i><font color="Green">E35</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font>ex <font color="Olive">b<sub>2</sub></font>, <font color="Olive">b<sub>3</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <br/>( <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>2</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Olive">b<sub>3</sub></font> )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>2</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>3</sub></font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>
<b>set </b><font color="Maroon">c<sub>4</sub></font> = the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>;<br/>
<a NAME="E3:49"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="orders_2.html#V3">transitive</a>
 
<b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dickson.html#D3" target="_self">Def3</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>5</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">c<sub>1</sub></font>;<br/>

<b>assume </b><a NAME="E4:49"/><i><font color="Green">E37</font></i>: 
(  not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> or  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1"/>
(  not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a> or  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E4:49"><i><font color="Green">E37</font></i></a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1"><b>suppose </b></a><a NAME="E1:49_1_1"/><i><font color="Green">E38</font></i>: 
not  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="finset_1.html#V1">finite</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> as  <a href="finset_1.html#V1">infinite</a>  <a href="hidden.html#M1">set</a>  ;<br/><b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font><b> such that </b><br/><a NAME="E4:49_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>7</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#T3" target="_self">Th3</a></i>;<br/><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/><div><i><font color="Green">E40</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:49_1_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="nat_1.html#NM1">Nat</a> <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E3:49_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font> is    <a href="subset_1.html#M1">Element</a> of  <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><a NAME="E4:49_1_1_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E5:49_1_1_1"/><b>then </b>
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E6:49_1_1_1"/><b>then </b>
 <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>9</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/><b>hence </b><a NAME="E7:49_1_1_1"/>
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">c<sub>8</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#E4:49_1_1_1"><i><font color="Green">E42</font></i></a></i>;<br/></div><b>end;</b></div><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> ,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="E7:49_1_1"/><i><font color="Green">E41</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a>  holds <br/><font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S2">FUNCT_2:sch 2</a>(<a class="txt" href="dickson.html#E5:49_1_1"><i><font color="Green">E40</font></i></a>);<br/></i><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> ;<br/><b>consider </b><font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>11</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E10:49_1_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font>
 <b>and </b><br/><a NAME="E11:49_1_1"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E2:49"><i><font color="Green">E35</font></i></a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font>, <font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> ;<br/><a NAME="E13:49_1_1"/><i><font color="Green">E44</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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="txt" href="dickson.html#E11:49_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E14:49_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E15:49_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E16:49_1_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font>
 ;<br/><a NAME="E17:49_1_1"/><b>then </b><i><font color="Green">E48</font></i>: 
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dickson.html#T23" target="_self">Th23</a></i>;<br/><a NAME="E18:49_1_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E19:49_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>9</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_1"><i><font color="Green">E41</font></i></a></i>;<br/><a NAME="E20:49_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E16:49_1_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E17:49_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E18:49_1_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="dickson.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E21:49_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E14:49_1_1"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E15:49_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E19:49_1_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="dickson.html#T20" target="_self">Th20</a></i>;<br/><a NAME="E22:49_1_1"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_1_2_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font> or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_1_2_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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> <a href="relset_1.html#K6">~</a> 
 <b>by </b><i><a class="txt" href="dickson.html#E13:49_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="relat_1.html#D7">RELAT_1:def 7</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:49_1_1_2_1_2"/><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>13</sub></font>
 ;<br/><div class="add"><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_1_2_1_2_1"/>
not <span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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> <a href="relset_1.html#K6">~</a> 
 ;<br/><a NAME="E2:49_1_1_2_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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> <a href="relset_1.html#K3">\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E13:49_1_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E3:49_1_1_2_1_2_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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 <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 ;<br/><b>hence </b><a NAME="E4:49_1_1_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E20:49_1_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="dickson.html#E22:49_1_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_1_2_1_2"><i><font color="Green">E54</font></i></a>, <a class="ref" href="waybel_4.html#D26">WAYBEL_4:def 26</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:49_1_1_2_1_2"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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> <a href="relset_1.html#K6">~</a> 
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><a NAME="E24:49_1_1"/>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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="txt" href="dickson.html#E11:49_1_1"><i><font color="Green">E43</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/><a NAME="E25:49_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></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> <a href="eqrel_1.html#K2">/\</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="relset_1.html#K6">~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E23:49_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E26:49_1_1"/><b>then </b>
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>12</sub></font>,<font color="Maroon">c<sub>13</sub></font></span><a href="yellow_3.html#K7">]</a></span> <a href="hidden.html#R2">in</a>  <a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="ref" href="dickson.html#D4" target="_self">Def4</a></i>;<br/><a NAME="E27:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T27">EQREL_1:27</a></i>;<br/><a NAME="E28:49_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>14</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E30:49_1_1"/>
<font color="Maroon">c<sub>14</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E31:49_1_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E16:49_1_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self">Def8</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>15</sub></font> being   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span><b> such that </b><br/><a NAME="E33:49_1_1"/>
<font color="Maroon">c<sub>15</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>and </b><br/><a NAME="E34:49_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E14:49_1_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="dickson.html#D8" target="_self">Def8</a></i>;<br/><a NAME="E35:49_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E17:49_1_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E18:49_1_1"><i><font color="Green">E49</font></i></a>, <a class="txt" href="dickson.html#E31:49_1_1"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E36:49_1_1"/><b>then </b><i><font color="Green">E57</font></i>: 
 <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>14</sub></font>
 <b>by </b><i><a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><a NAME="E37:49_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>15</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E15:49_1_1"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E19:49_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="dickson.html#E34:49_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E38:49_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>11</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E28:49_1_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="dickson.html#E31:49_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E34:49_1_1"><i><font color="Green">E56</font></i></a>, <a class="txt" href="dickson.html#E36:49_1_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="eqrel_1.html#T31">EQREL_1:31</a></i>;<br/><b>hence </b><a NAME="E39:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="dickson.html#E4:49_1_1"><i><font color="Green">E39</font></i></a>, <a class="txt" href="dickson.html#E10:49_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="funct_2.html#T25">FUNCT_2:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2"><b>suppose </b></a><a NAME="E1:49_1_2"/><i><font color="Green">E38</font></i>: 
 <a href="dickson.html#K6" target="_self">min-classes</a> <font color="Maroon">c<sub>5</sub></font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1">Element</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span>)</span> =  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">a<sub>2</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">a<sub>2</sub></font></span>)</span></span>)</span>;<br/><b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="funct_1.html#NM1">Function</a><b> such that </b><br/><a NAME="E3:49_1_2"/><i><font color="Green">E39</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="numbers.html#K5">NAT</a> 
 <b>and </b><br/><a NAME="E4:49_1_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <font color="Maroon">c<sub>5</sub></font>
 <b>and </b><br/><a NAME="E5:49_1_2"/><i><font color="Green">E41</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">c<sub>6</sub></font> <a href="funct_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> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font>)
 <b>from </b><i><a class="ref" href="recdef_1.html#S3">RECDEF_1:sch 3</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>;<br/><a NAME="E6:49_1_2"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="dickson.html#E4:49_1_2"><i><font color="Green">E40</font></i></a></i>;<br/><div><i><font color="Green">E43</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>assume </b><a NAME="E1:49_1_2_1"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font>]
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span>;<br/><a NAME="E3:49_1_2_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <span class="p4">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:49_1_2_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>8</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_1_1"/><b>then </b>
<font color="Maroon">c<sub>8</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_1"><i><font color="Green">E44</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><a NAME="E5:49_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <span class="p2">(<span class="default"><font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font></span>)</span></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_1"><i><font color="Green">E45</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E6:49_1_2_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>7</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/></div><b>end;</b></div><a NAME="E8:49_1_2"/><i><font color="Green">E44</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E6:49_1_2"><i><font color="Green">E42</font></i></a>, <a class="txt" href="dickson.html#E7:49_1_2"><i><font color="Green">E43</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_2"><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="E1:49_1_2_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 ;<br/><a NAME="E2:49_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green">E44</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_2"><i><font color="Green">E45</font></i></a></i>;<br/><b>hence </b><a NAME="E3:49_1_2_2"/>
<font color="Maroon">c<sub>6</sub></font> <a href="funct_1.html#K1">.</a> <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>
 ;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">c<sub>7</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">c<sub>1</sub></font> <b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_2.html#T5">FUNCT_2:5</a></i>;<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E1:49_1_2_3"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:49_1_2_3"/><i><font color="Green">E47</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49_1_2_3_1"/><i><font color="Green">E48</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> implies <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:49_1_2_3_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:49_1_2_3_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:49_1_2_3_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> ;<br/>
<b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/>
<a NAME="E5:49_1_2_3_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E6:49_1_2_3_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green">E41</font></i></a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_3_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_3_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:49_1_2_3_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_3_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_3_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div>
<a NAME="E8:49_1_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E6:49_1_2_3_1"><i><font color="Green">E52</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E9:49_1_2_3_1"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <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> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E10:49_1_2_3_1"/><b>then </b><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="yellow_3.html#K7">[</a><span class="default"><font color="Maroon">c<sub>11</sub></font>,<font color="Maroon">c<sub>10</sub></font></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="wellord1.html#D1">WELLORD1:def 1</a></i>;<br/>
<a NAME="E11:49_1_2_3_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_2_3_1_2"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_3_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2_1"><b>suppose </b></a><a NAME="E1:49_1_2_3_1_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_3_1_2_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_3_1"><i><font color="Green">E48</font></i></a>, <a class="txt" href="dickson.html#E11:49_1_2_3_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_3_1_2_2"><b>suppose </b></a><a NAME="E1:49_1_2_3_1_2_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:49_1_2_3_1_2_2"/>
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E10:49_1_2_3_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_2.html#D9">ORDERS_2:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:49_1_2_3"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E1:49_1_2_3"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_3"><i><font color="Green">E47</font></i></a>);</i><br/></div><b>end;</b></div><div><i><font color="Green">E46</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[  <a href="nat_1.html#NM1">Nat</a>] means ( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">a<sub>1</sub></font> implies  not <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E1:49_1_2_4"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="nat_1.html#T18">NAT_1:18</a></i>;<br/><a NAME="E2:49_1_2_4"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>] holds <br/><font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:49_1_2_4_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> implies  not <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>and </b><br/><a NAME="E2:49_1_2_4_1"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<a NAME="E3:49_1_2_4_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E2:49_1_2_4_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>, <font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span> ;<br/>
<a NAME="E5:49_1_2_4_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font>
 
<b>by </b><i><a class="txt" href="dickson.html#E8:49_1_2"><i><font color="Green">E44</font></i></a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:49_1_2_4_1_1"/>
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> or <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> )
 </span><b>by </b><i><a class="txt" href="dickson.html#E3:49_1_2_4_1"><i><font color="Green">E50</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1_1"><b>suppose </b></a><a NAME="E1:49_1_2_4_1_1_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:49_1_2_4_1_1_1"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E3:49_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:49_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:49_1_2_4_1_1_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green">E45</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_4_1_1_1"><i><font color="Green">E52</font></i></a></i>;<br/><a NAME="E6:49_1_2_4_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E3:49"><i><font color="Green">E36</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4_1_1_1"><i><font color="Green">E53</font></i></a>, <a class="ref" href="orders_2.html#T26">ORDERS_2:26</a></i>;<br/><a NAME="E7:49_1_2_4_1_1_1"/><b>then </b><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E4:49_1_2_4_1_1_1"><i><font color="Green">E54</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/><a NAME="E8:49_1_2_4_1_1_1"/><i><font color="Green">E56</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_4_1_1_1_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:49_1_2_4_1_1_1_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_4_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_4_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_4_1_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E10:49_1_2_4_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E7:49_1_2_4_1_1_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="dickson.html#E8:49_1_2_4_1_1_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="49_1_2_4_1_1_2"><b>suppose </b></a><a NAME="E1:49_1_2_4_1_1_2"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font>
 ;<br/><div class="add"><b>assume </b><a NAME="E2:49_1_2_4_1_1_2"/><i><font color="Green">E53</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/><a NAME="E3:49_1_2_4_1_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E4:49_1_2_4_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>9</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E11:49_1_2"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E5:49_1_2_4_1_1_2"/><b>then </b><i><font color="Green">E54</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E1:49_1_2_4_1_1_2"><i><font color="Green">E52</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4_1_1_2"><i><font color="Green">E53</font></i></a>, <a class="ref" href="dickson.html#T9" target="_self">Th9</a></i>;<br/><b>set </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span>;<br/><a NAME="E6:49_1_2_4_1_1_2"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="subset_1.html#K10">choose</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p2">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p3">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2"><i><font color="Green">E41</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_2_4_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:49_1_2_4_1_1_2_1"/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><a href="eqrel_1.html#K7">Class</a> <span class="p2">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font></span>)</span> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><a NAME="E2:49_1_2_4_1_1_2_1"/><b>then </b>
<span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">c<sub>1</sub></font> <a href="wellord1.html#K1">-Seg</a> <font color="Maroon">c<sub>10</sub></font></span>)</span> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="dickson.html#K1" target="_self">EqRel</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>,<font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:49_1_2_4_1_1_2_1"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="waybel_4.html#R4">is_minimal_wrt</a> <font color="Maroon">c<sub>5</sub></font>,the <a href="orders_2.html#U1">InternalRel</a> of <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="dickson.html#K5" target="_self">\~</a> </span>)</span>
 <b>by </b><i><a class="txt" href="dickson.html#E1:49"><i><font color="Green">E34</font></i></a>, <a class="txt" href="dickson.html#E5:49_1_2_4_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="dickson.html#T31" target="_self">Th31</a></i>;<br/><b>hence </b><a NAME="E4:49_1_2_4_1_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E1:49_1_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="dickson.html#T22" target="_self">Th22</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E8:49_1_2_4_1_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E5:49_1_2_4_1_1_2"><i><font color="Green">E54</font></i></a>, <a class="txt" href="dickson.html#E6:49_1_2_4_1_1_2"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>thus </b><a NAME="E3:49_1_2_4"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="dickson.html#E1:49_1_2_4"><i><font color="Green">E47</font></i></a>, <a class="txt" href="dickson.html#E2:49_1_2_4"><i><font color="Green">E48</font></i></a>);</i><br/></div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>9</sub></font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E14:49_1_2"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="orders_2.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">c<sub>9</sub></font> )
 <b>by </b><i><a class="txt" href="dickson.html#E2:49"><i><font color="Green">E35</font></i></a></i>;<br/><b>thus </b><a NAME="E15:49_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="dickson.html#E12:49_1_2"><i><font color="Green">E46</font></i></a>, <a class="txt" href="dickson.html#E14:49_1_2"><i><font color="Green">E47</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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