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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>, <font color="Maroon">c<sub>4</sub></font> be   <a href="supinf_1.html#NM3">R_eal</a>;<br/>







<b>assume </b><b>that </b><br/><a NAME="E1:60"/><i><font color="Green">E38</font></i>: 
( not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>and </b><br/><a NAME="E2:60"/><i><font color="Green">E39</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>and </b><br/><a NAME="E3:60"/><i><font color="Green">E40</font></i>: 
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/>

<a NAME="E4:60"/>
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="extreal1.html#K4">.|</a></span> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><font color="Maroon">c<sub>2</sub></font></span><a href="extreal1.html#K4">.|</a></span> )
 
<b>by </b><i><a class="ref" href="extreal2.html#T51" target="_self">Th51</a></i>;<br/>
<a NAME="E5:60"/><b>then </b><i><font color="Green">E41</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 0 )
 
<b>by </b><i><a class="txt" href="extreal2.html#E2:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="extreal2.html#E3:60"><i><font color="Green">E40</font></i></a></i>;<br/>
<a NAME="E6:60"/><b>then </b>
(  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>4</sub></font> )
 
;<br/>
<a NAME="E7:60"/><b>then </b>
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_2.html#K3">-</a> <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#T17">SUPINF_2:17</a></i>;<br/>
<a NAME="E8:60"/><b>then </b><i><font color="Green">E42</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="extreal1.html#T4">EXTREAL1:4</a></i>;<br/>
<a NAME="E9:60"/><i><font color="Green">E43</font></i>: 
(  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp;  <a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>4</sub></font> )
 
<b>by </b><i><a class="txt" href="extreal2.html#E2:60"><i><font color="Green">E39</font></i></a>, <a class="txt" href="extreal2.html#E3:60"><i><font color="Green">E40</font></i></a>, <a class="ref" href="extreal2.html#T60" target="_self">Th60</a></i>;<br/>
<a NAME="E10:60"/><b>then </b><i><font color="Green">E44</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="extreal2.html#E1:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="extreal2.html#E5:60"><i><font color="Green">E41</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/>
<a NAME="E11:60"/><i><font color="Green">E45</font></i>: 
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="supinf_2.html#K2">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="extreal2.html#E1:60"><i><font color="Green">E38</font></i></a>, <a class="txt" href="extreal2.html#E8:60"><i><font color="Green">E42</font></i></a>, <a class="txt" href="extreal2.html#E9:60"><i><font color="Green">E43</font></i></a>, <a class="ref" href="supinf_2.html#T14">SUPINF_2:14</a></i>;<br/>
<a NAME="E12:60"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K3">-</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_2.html#K3">-</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="extreal2.html#T14" target="_self">Th14</a></i>;<br/>
<b>hence </b><a NAME="E13:60"/>
<span class="p1"><a href="extreal1.html#K4">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="extreal1.html#K4">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="extreal2.html#E10:60"><i><font color="Green">E44</font></i></a>, <a class="txt" href="extreal2.html#E11:60"><i><font color="Green">E45</font></i></a>, <a class="ref" href="extreal2.html#T60" target="_self">Th60</a></i>;<br/>


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