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<div class="add">

<b>let </b><font color="Maroon">X</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">Y</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> , <font color="Maroon">Z</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="E1:48"/><i><font color="Green">E24</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K1">+infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> &amp;  <a href="supinf_1.html#K2">-infty</a>  <a href="hidden.html#R2">in</a> <font color="Maroon">Z</font> ) )
 ;<br/>

<b>let </b><font color="Maroon">F</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Y</font>;<br/><b>let </b><font color="Maroon">G</font> be   <a href="funct_2.html#NM1">Function</a> of <font color="Maroon">X</font>,<font color="Maroon">Z</font>;<br/>



<b>assume </b><a NAME="E2:48"/><i><font color="Green">E28</font></i>: 
( not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">G</font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 ;<br/>

<a NAME="E3:48"/>
 <a href="supinf_2.html#K7" target="_self">rng</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</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">F</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">G</font></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T7" target="_self">Th7</a></i>;<br/>
<a NAME="E4:48"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="supinf_2.html#K8" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K6">sup</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">F</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">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="supinf_1.html#T91">SUPINF_1:91</a></i>;<br/>
<a NAME="E5:48"/>
( 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">F</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">G</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">F</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">G</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</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#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</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="48_1"><b>proof </b></a><div class="add">

<a NAME="E1:48_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">F</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">G</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">F</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">G</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:48_1_1"/><i><font color="Green">E30</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">F</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">G</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">F</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">G</font> ) )
 ;<br/>

<a NAME="E2:48_1_1"/><i><font color="Green">E31</font></i>: 
 <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<a NAME="E3:48_1_1"/>
 <a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">Z</font>
 
<b>by </b><i><a class="ref" href="relset_1.html#T12">RELSET_1:12</a></i>;<br/>
<b>hence </b><a NAME="E4:48_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="supinf_2.html#T11" target="_self">Th11</a>, <a class="ref" href="supinf_2.html#T12" target="_self">Th12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:48_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">F</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">G</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">F</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">G</font> ) &amp; not <!--HUMANRECFAILED-->(  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</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#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp;  <a href="supinf_1.html#K6">sup</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  ) )
 <b>by </b><i><a class="ref" href="supinf_2.html#T9" target="_self">Th9</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:48"/><b>then </b>
 <a href="supinf_1.html#K6">sup</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">F</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">G</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_1.html#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">F</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#K6">sup</a> <span class="p2">(<span class="default"><a href="supinf_2.html#K7" target="_self">rng</a> <font color="Maroon">G</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="supinf_2.html#E4:48"><i><font color="Green">E29</font></i></a></i>;<br/>
<b>hence </b><a NAME="E7:48"/>
 <a href="supinf_2.html#K8" target="_self">sup</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="supinf_2.html#K11" target="_self">+</a> <font color="Maroon">G</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">F</font></span>)</span> <a href="supinf_2.html#K2" target="_self">+</a> <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">G</font></span>)</span>
 <b>by </b><i><a class="ref" href="supinf_2.html#T10" target="_self">Th10</a>, <a class="ref" href="supinf_1.html#T29">SUPINF_1:29</a></i>;<br/>


</div>
