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<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:42"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 <b>and </b><br/><a NAME="E2:42"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K1">0.</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 ;<br/>

<a NAME="E3:42"/><i><font color="Green">E26</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="txt" href="extreal1.html#E1:42"><i><font color="Green">E24</font></i></a>, <a class="ref" href="supinf_1.html#T20">SUPINF_1:20</a>, <a class="ref" href="supinf_1.html#T21">SUPINF_1:21</a></i>;<br/>
<a NAME="E4:42"/><i><font color="Green">E27</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#T7" target="_self">Th7</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> <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1"/>
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a>  or <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1"><b>suppose </b></a><a NAME="E1:42_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</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="extreal1.html#E3:42"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_1"/><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self">Th32</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_1_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_1_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E3:42"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_1_1_1_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self">Th32</a></i>;<br/><a NAME="E4:42_1_1_1_1_1"/>
( <font color="Maroon">c<sub>6</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>5</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;</a> 0 )
 <b>by </b><i><a class="txt" href="extreal1.html#E1:42"><i><font color="Green">E24</font></i></a>, <a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/><a NAME="E5:42_1_1_1_1_1"/><b>then </b>
<font color="Maroon">c<sub>6</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="real_1.html#T74">REAL_1:74</a></i>;<br/><b>hence </b><a NAME="E6:42_1_1_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:42_1_1"><i><font color="Green">E28</font></i></a>, <a class="txt" href="extreal1.html#E3:42_1_1_1_1_1"><i><font color="Green">E29</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_1_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_1_1_1_2"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E4:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self">Def2</a></i>;<br/><b>hence </b><a NAME="E3:42_1_1_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E3:42_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E5:42_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2"><b>suppose </b></a><a NAME="E1:42_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/><div class="add"><a NAME="E2:42_1_2"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E4:42"><i><font color="Green">E27</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self">Def2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1_2_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:42_1_2_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="extreal1.html#E3:42"><i><font color="Green">E26</font></i></a>, <a class="ref" href="xxreal_0.html#T14">XXREAL_0:14</a></i>;<br/><a NAME="E3:42_1_2_1_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>5</sub></font> <a href="real_1.html#K6">/</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="ref" href="extreal1.html#T32" target="_self">Th32</a></i>;<br/><b>hence </b><a NAME="E4:42_1_2_1_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="42_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:42_1_2_1_1_2"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a> 
 ;<br/><div class="add"><b>hence </b><a NAME="E2:42_1_2_1_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="extreal1.html#E2:42"><i><font color="Green">E25</font></i></a>, <a class="txt" href="extreal1.html#E4:42"><i><font color="Green">E27</font></i></a>, <a class="txt" href="extreal1.html#E2:42_1_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="extreal1.html#D2" target="_self">Def2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E4:42_1_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>1</sub></font> <a href="extreal1.html#K3" target="_self">/</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

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