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<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<b>assume </b><a NAME="E1:2"/><i><font color="Green">E3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>1</sub></font>]
 ;<br/>

<b>let </b><font color="Maroon">c<sub>2</sub></font> be  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><a NAME="E2:2"/><i><font color="Green">E4</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp;  <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:2_1_1_1"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="card_2.html#T60">CARD_2:60</a></i>;<br/><a NAME="E4:2_1_1_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/><a NAME="E6:2_1_1_1"/>
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span> )
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E7:2_1_1_1"/><b>then </b><i><font color="Green">E6</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V3">bounded</a>
 <b>by </b><i><a class="ref" href="seq_4.html#T15">SEQ_4:15</a></i>;<br/><b>hence </b><a NAME="E8:2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><a NAME="E9:2_1_1_1"/>
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/><b>hence </b><a NAME="E10:2_1_1_1"/>
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><b>thus </b><a NAME="E11:2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="txt" href="goboard2.html#E7:2_1_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="seq_4.html#D3">SEQ_4:def 3</a></i>;<br/><a NAME="E12:2_1_1_1"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="seq_4.html#T22">SEQ_4:22</a></i>;<br/><b>hence </b><a NAME="E13:2_1_1_1"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E3:2_1_1_1"><i><font color="Green">E5</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2"/><i><font color="Green">E5</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><div class="add"><b>consider </b><font color="Maroon">c<sub>3</sub></font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">c<sub>2</sub></font>;<br/><b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span> as  <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K1">REAL</a>  ;<br/><a NAME="E5:2_1_1_2"/>
<span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><a NAME="E6:2_1_1_2"/><b>then </b>
 <a href="card_1.html#K4">card</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>2</sub></font> <a href="subset_1.html#K7">\</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="real_1.html#K5">-</a> <span class="p1">(<span class="default"><a href="card_1.html#K4">card</a> <span class="p2"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="card_2.html#T63">CARD_2:63</a></i>;<br/><b>then </b> <a href="card_1.html#K4">card</a> <font color="Maroon">c<sub>5</sub></font> = 
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="real_1.html#K5">-</a> 1
<b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a>, <a class="ref" href="card_1.html#T79">CARD_1:79</a></i>
<br/>.= 
<font color="Maroon">c<sub>1</sub></font>

;<br/><a NAME="E8:2_1_1_2"/><b>then </b><i><font color="Green">E6</font></i>: 
( <font color="Maroon">c<sub>5</sub></font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="seq_4.html#V2">bounded_below</a> &amp;  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</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="goboard2.html#E1:2"><i><font color="Green">E3</font></i></a>, <a class="txt" href="goboard2.html#E1:2_1_1_2"><i><font color="Green">E5</font></i></a>, <a class="ref" href="card_1.html#T78">CARD_1:78</a></i>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> =  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>7</sub></font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>8</sub></font> =  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font>;<br/><b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>;<br/><a NAME="E9:2_1_1_2"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> &amp;  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font> )
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a>, <a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/><div><i><font color="Green">E8</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_2_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_2_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_2_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T25">XXREAL_0:25</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_2_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_2_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:2_1_1_2_2_1_1_2"/><b>then </b>
<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="goboard2.html#E1:2_1_1_2_2"><i><font color="Green">E9</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:2_1_1_2_2_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E5:2_1_1_2_2_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="goboard2.html#E9:2_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_2_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E11:2_1_1_2"/><i><font color="Green">E9</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D1">SEQ_4:def 1</a></i>;<br/><div><i><font color="Green">E10</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_3"/><i><font color="Green">E11</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_3_1_1"/>
(  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_3_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_3_1_1_1"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_3_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_3_1_1_1"/>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:2_1_1_2_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_3_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_3_1_1_2"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_3_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_3_1_1_2"/>
<span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:2_1_1_2_3"><i><font color="Green">E11</font></i></a>, <a class="ref" href="xreal_1.html#T46">XREAL_1:46</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:2_1_1_2_3"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <span class="p1">(<span class="default"><a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_4_1"/>
(  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T16">XXREAL_0:16</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_4_1_1"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_4_1_1"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_4_1_2"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_4_1_2"/>
 <a href="xxreal_0.html#K4">max</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</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>hence </b><a NAME="E14:2_1_1_2"/>
 <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E10:2_1_1_2"><i><font color="Green">E8</font></i></a>, <a class="txt" href="goboard2.html#E11:2_1_1_2"><i><font color="Green">E9</font></i></a>, <a class="txt" href="goboard2.html#E12:2_1_1_2"><i><font color="Green">E10</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><div><i><font color="Green">E11</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_5"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_5_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_5_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_5_1_1"/>
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_5_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_5_1_1_1"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_5_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="ref" href="xxreal_0.html#T17">XXREAL_0:17</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_5_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_5_1_1_2"/>
<font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><a NAME="E2:2_1_1_2_5_1_1_2"/><b>then </b>
not <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>4</sub></font></span><a href="seq_4.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:2_1_1_2_5_1_1_2"/><b>then </b>
<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="goboard2.html#E1:2_1_1_2_5"><i><font color="Green">E12</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E4:2_1_1_2_5_1_1_2"/><b>then </b>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E5:2_1_1_2_5_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E9:2_1_1_2"><i><font color="Green">E7</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E3:2_1_1_2_5"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E16:2_1_1_2"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 <b>by </b><i><a class="ref" href="seq_4.html#D2">SEQ_4:def 2</a></i>;<br/><div><i><font color="Green">E13</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_6"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>10</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:2_1_1_2_6"/><i><font color="Green">E14</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font>
 ;<br/><b>reconsider </b><font color="Maroon">c<sub>11</sub></font> =  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> as  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  ;<br/><b>take </b><font color="Maroon">c<sub>12</sub></font> = <font color="Maroon">c<sub>11</sub></font>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_6_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_6_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_6_1_1"/>
(  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_6_1_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_6_1_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_6_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_6_1_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:2_1_1_2_6"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_6_1_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_6_1_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_6_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</font></i></a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>thus </b><a NAME="E3:2_1_1_2_6_1_1_2"/>
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E1:2_1_1_2_6"><i><font color="Green">E14</font></i></a>, <a class="ref" href="real_2.html#T174">REAL_2:174</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:2_1_1_2_6"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p2">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>10</sub></font> )
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="2_1_1_2_7"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:2_1_1_2_7_1"/>
(  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> or  <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font> )
 </span><b>by </b><i><a class="ref" href="xxreal_0.html#T15">XXREAL_0:15</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_1"><b>suppose </b></a><a NAME="E1:2_1_1_2_7_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_7_1_1"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E2:2"><i><font color="Green">E4</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="2_1_1_2_7_1_2"><b>suppose </b></a><a NAME="E1:2_1_1_2_7_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:2_1_1_2_7_1_2"/>
 <a href="xxreal_0.html#K3">min</a> <font color="Maroon">c<sub>4</sub></font>,<span class="p1">(<span class="default"><a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E8:2_1_1_2"><i><font color="Green">E6</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>hence </b><a NAME="E19:2_1_1_2"/>
 <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="goboard2.html#E15:2_1_1_2"><i><font color="Green">E11</font></i></a>, <a class="txt" href="goboard2.html#E16:2_1_1_2"><i><font color="Green">E12</font></i></a>, <a class="txt" href="goboard2.html#E17:2_1_1_2"><i><font color="Green">E13</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E4:2"/>
( <font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V1">bounded_above</a> &amp;  <a href="seq_4.html#K4">upper_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a> &amp;  <a href="seq_4.html#K5">lower_bound</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/>


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