<?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="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</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="E1:49"/><i><font color="Green">E39</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>3</sub></font> ) )
 ;<br/>

<b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>;<br/>



<b>assume </b><a NAME="E2:49"/><i><font color="Green">E40</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 ;<br/>

<a NAME="E3:49"/>
 <a href="supinf_2.html#K7" target="_self">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T34" target="_self">Th34</a></i>;<br/>
<a NAME="E4:49"/><b>then </b><i><font color="Green">E41</font></i>: 
 <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K9" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T94">SUPINF_1:94</a></i>;<br/>
<a NAME="E5:49"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1"><b>proof </b></a><div class="add">

<a NAME="E1:49_1"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="49_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:49_1_1"/><i><font color="Green">E42</font></i>: 
( (  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) or (  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) )
 ;<br/>

<a NAME="E2:49_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:49_1_1"/>
 <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:49_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E1:49"><i><font color="Green">E39</font></i></a>, <a class="txt" href="supinf_2.html#E1:49_1_1"><i><font color="Green">E42</font></i></a>, <a class="txt" href="supinf_2.html#E2:49_1_1"><i><font color="Green">E43</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:49_1"/>
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a>  <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>by </b><i><a class="txt" href="supinf_2.html#E2:49"><i><font color="Green">E40</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:49"/><b>then </b>
<span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K7">inf</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K7">inf</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K5" target="_self">+</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">c<sub>5</sub></font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T27" target="_self">Th27</a></i>;<br/>
<b>hence </b><a NAME="E7:49"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K9" target="_self">inf</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_2.html#K9" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>4</sub></font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">c<sub>5</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="supinf_2.html#E4:49"><i><font color="Green">E41</font></i></a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


</div>
