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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of <a href="numbers.html#K6">ExtREAL</a> ;<br/>

<b>assume </b><a NAME="E1:41"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> is <a href="supinf_1.html#V4" target="_self">bounded_below</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K6">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>2</sub></font> being    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font><b> such that </b><br/><a NAME="E3:41"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="ref" href="supinf_1.html#D12" target="_self">Def12</a></i>;<br/>
<a NAME="E4:41"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:41"/><i><font color="Green">E30</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> &amp; not <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41"><i><font color="Green">E27</font></i></a>, <a class="ref" href="supinf_1.html#T60" target="_self">Th60</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <a href="supinf_1.html#K3" target="_self">REAL</a>  as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3" target="_self">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <a href="supinf_1.html#K3" target="_self">REAL</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> as   <a href="subset_1.html#NM2">Subset</a> of <a href="supinf_1.html#K3" target="_self">REAL</a>  <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E3:41"><i><font color="Green">E28</font></i></a></i>;<br/>
<a NAME="E10:41"/><i><font color="Green">E31</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:41"><i><font color="Green">E30</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>7</sub></font>, <font color="Maroon">c<sub>8</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="E1:41_1"/><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/><a NAME="E2:41_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:41_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_1"><i><font color="Green">E32</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E4:41_1"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>8</sub></font> is   <a href="real_1.html#NM1">Real</a> &amp; <font color="Maroon">c<sub>7</sub></font> is   <a href="real_1.html#NM1">Real</a> )
 <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E6:41_1"/><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>9</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="supinf_1.html#E3:41_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/><b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/><a NAME="E8:41_1"/>
<font color="Maroon">c<sub>9</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="supinf_1.html#E2:41_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="supinf_1.html#E6:41_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/><a NAME="E9:41_1"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <br/>( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>9</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E4:41_1"><i><font color="Green">E35</font></i></a></i>;<br/><b>hence </b><a NAME="E10:41_1"/>
<font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font>
 ;<br/></div><b>end;</b></div>
<b>then consider </b><font color="Maroon">c<sub>7</sub></font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E13:41"/><i><font color="Green">E32</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>7</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="axioms.html#T26">AXIOMS:26</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>7</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>9</sub></font> = <font color="Maroon">c<sub>8</sub></font>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>8</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E16:41"/><i><font color="Green">E33</font></i>: 
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/>( <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>11</sub></font>, <font color="Maroon">c<sub>12</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E1:41_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> )
 ;<br/>

<a NAME="E2:41_2"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_1_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_1_1_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:41_2_1_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E4:41_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_1_1_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_1.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:41_2_1_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:41_2_1_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="supinf_1.html#E1:41_2_1_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:41_2_1"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E5:41_2_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_1_2"/>
<font color="Maroon">c<sub>6</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="supinf_1.html#E4:41"><i><font color="Green">E29</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>hence </b><a NAME="E2:41_2_1_2"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E13:41"><i><font color="Green">E32</font></i></a>, <a class="txt" href="supinf_1.html#E4:41_2_1"><i><font color="Green">E37</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E6:41_2_1"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:41_2"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_2"/><i><font color="Green">E37</font></i>: 
( not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_2"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E4:41_2_2"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:41"><i><font color="Green">E30</font></i></a>, <a class="txt" href="supinf_1.html#E2:41_2_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E5:41_2_2"/><b>then </b><i><font color="Green">E39</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E6:41_2_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="supinf_1.html#E5:41_2_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E8:41_2_2"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>14</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="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>13</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E10:41_2_2"/>
( <font color="Maroon">c<sub>14</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>15</sub></font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E10:41"><i><font color="Green">E31</font></i></a>, <a class="txt" href="supinf_1.html#E13:41"><i><font color="Green">E32</font></i></a>, <a class="txt" href="supinf_1.html#E8:41_2_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E11:41_2_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xxreal_0.html#T4">XXREAL_0:4</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:41_2"/>
( not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; not <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  implies ( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_3"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a>  &amp; <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1" target="_self">+infty</a>  )
 ;<br/>

<a NAME="E2:41_2_3"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3_1"><b>proof </b></a><div class="add">

<a NAME="E1:41_2_3_1"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_2_3_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:41_2_3_1_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2" target="_self">-infty</a> 
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>6</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<a NAME="E3:41_2_3_1_1"/>
<font color="Maroon">c<sub>13</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<b>hence </b><a NAME="E4:41_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_3_1_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="supinf_1.html#T17" target="_self">Th17</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:41_2_3_1"/>
( <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:41_2_3_1"/>
<font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a> 
 <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="supinf_1.html#E1:41_2_3_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">c<sub>13</sub></font> = <font color="Maroon">c<sub>12</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E4:41_2_3"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>14</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> <b>by </b><i><a class="ref" href="xxreal_0.html#D1">XXREAL_0:def 1</a></i>;<br/>
<a NAME="E6:41_2_3"/>
<font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>14</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E6:41"><i><font color="Green">E30</font></i></a>, <a class="txt" href="supinf_1.html#E4:41_2_3"><i><font color="Green">E38</font></i></a>, <a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>
<a NAME="E7:41_2_3"/><b>then </b><i><font color="Green">E39</font></i>: 
not <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1" target="_self">+infty</a> 
 
<b>by </b><i><a class="ref" href="supinf_1.html#T18" target="_self">Th18</a></i>;<br/>
<a NAME="E8:41_2_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3" target="_self">REAL</a>  or <font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2" target="_self">-infty</a> ,<a href="supinf_1.html#K1" target="_self">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>15</sub></font> = <font color="Maroon">c<sub>11</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2_3"><i><font color="Green">E37</font></i></a>, <a class="txt" href="supinf_1.html#E7:41_2_3"><i><font color="Green">E39</font></i></a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>
<a NAME="E10:41_2_3"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c<sub>15</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="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:41_2_3"/>
<font color="Maroon">c<sub>13</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E1:41_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E12:41_2_3"/><b>then </b>
( <font color="Maroon">c<sub>15</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>13</sub></font> )
 
<b>by </b><i><a class="txt" href="supinf_1.html#E13:41"><i><font color="Green">E32</font></i></a>, <a class="txt" href="supinf_1.html#E10:41_2_3"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:41_2_3"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E5:41_2"/>
( <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> )
 <b>by </b><i><a class="txt" href="supinf_1.html#E2:41_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="supinf_1.html#E3:41_2"><i><font color="Green">E36</font></i></a>, <a class="ref" href="xxreal_0.html#T3">XXREAL_0:3</a>, <a class="ref" href="xxreal_0.html#T5">XXREAL_0:5</a></i>;<br/>


</div><b>end;</b></div>
<b>take </b>
<font color="Maroon">c<sub>10</sub></font>
;<br/>

<a NAME="E17:41"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="supinf_1.html#NM3">R_eal</a> st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>1</sub></font>
 
<b>by </b><i><a class="txt" href="supinf_1.html#E4:41"><i><font color="Green">E29</font></i></a>, <a class="txt" href="supinf_1.html#E16:41"><i><font color="Green">E33</font></i></a></i>;<br/>
<b>hence </b><a NAME="E18:41"/>
<font color="Maroon">c<sub>10</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="supinf_1.html#D10" target="_self">Def10</a></i>;<br/>

<b>let </b><font color="Maroon">c<sub>11</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>

<b>assume </b><a NAME="E19:41"/>
<font color="Maroon">c<sub>11</sub></font> is    <a href="supinf_1.html#M2" target="_self">minorant</a> of <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<a NAME="E20:41"/><b>then </b>
<font color="Maroon">c<sub>11</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K5" target="_self">SetMinorant</a> <font color="Maroon">c<sub>1</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_1.html#D15" target="_self">Def15</a></i>;<br/>
<b>hence </b><a NAME="E21:41"/>
<font color="Maroon">c<sub>11</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="supinf_1.html#E6:41"><i><font color="Green">E30</font></i></a>, <a class="txt" href="supinf_1.html#E16:41"><i><font color="Green">E33</font></i></a></i>;<br/>


</div>
