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<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="E1:65"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a>  &amp; <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<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="E2:65"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="supinf_2.html#V1" target="_self">bounded_above</a> )
 ;<br/>

<a NAME="E3:65"/><i><font color="Green">E53</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> ) )
 
<b>by </b><i><a class="txt" href="supinf_2.html#E1:65"><i><font color="Green">E51</font></i></a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a></i>;<br/>
<a NAME="E4:65"/><i><font color="Green">E54</font></i>: 
( not  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; not  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <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#E2:65"><i><font color="Green">E52</font></i></a>, <a class="ref" href="supinf_2.html#D11" target="_self">Def11</a></i>;<br/>
<a NAME="E5:65"/><i><font color="Green">E55</font></i>: 
(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_1"/><i><font color="Green">E56</font></i>: 
(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  )
 ;<br/>

<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> =  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font>, <font color="Maroon">c<sub>7</sub></font> =  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> as   <a href="supinf_1.html#NM3">R_eal</a> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>6</sub></font>, <font color="Maroon">c<sub>9</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="txt" href="supinf_2.html#E1:65_1"><i><font color="Green">E56</font></i></a></i>;<br/>
<a NAME="E4:65_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>9</sub></font>
 
<b>by </b><i><a class="ref" href="supinf_2.html#T1" target="_self">Th1</a></i>;<br/>
<b>hence </b><a NAME="E5:65_1"/>
<span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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="supinf_1.html#T31">SUPINF_1:31</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:65"/><i><font color="Green">E56</font></i>: 
(  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  &amp;  <a href="supinf_2.html#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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="supinf_2.html#D2" target="_self">Def2</a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="xxreal_0.html#T7">XXREAL_0:7</a></i>;<br/>
<a NAME="E7:65"/><i><font color="Green">E57</font></i>: 
(  <a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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="supinf_2.html#D2" target="_self">Def2</a>, <a class="ref" href="supinf_1.html#T1">SUPINF_1:1</a>, <a class="ref" href="xxreal_0.html#T7">XXREAL_0:7</a></i>;<br/>
<a NAME="E8:65"/><i><font color="Green">E58</font></i>: 
(  <a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  implies <span class="p1">(<span class="default"><a href="supinf_2.html#K8" target="_self">sup</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#K8" target="_self">sup</a> <font color="Maroon">c<sub>5</sub></font></span>)</span> <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="supinf_2.html#D2" target="_self">Def2</a>, <a class="ref" href="xxreal_0.html#T7">XXREAL_0:7</a></i>;<br/>
<a NAME="E9:65"/>
 <a href="supinf_2.html#K8" target="_self">sup</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="65_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:65_2"/>
not  <a href="supinf_2.html#K8" target="_self">sup</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="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:65_2"/><b>then </b>
(  not  <a href="supinf_2.html#K8" target="_self">sup</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="xxreal_0.html#R1">&lt;=</a>  <a href="supinf_1.html#K1">+infty</a>  or  <a href="supinf_2.html#K8" target="_self">sup</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="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_1.html#T22">SUPINF_1:22</a></i>;<br/>
<b>hence </b><a NAME="E3:65_2"/>
contradiction
 <b>by </b><i><a class="txt" href="supinf_2.html#E3:65"><i><font color="Green">E53</font></i></a>, <a class="txt" href="supinf_2.html#E4:65"><i><font color="Green">E54</font></i></a>, <a class="txt" href="supinf_2.html#E5:65"><i><font color="Green">E55</font></i></a>, <a class="txt" href="supinf_2.html#E6:65"><i><font color="Green">E56</font></i></a>, <a class="txt" href="supinf_2.html#E7:65"><i><font color="Green">E57</font></i></a>, <a class="txt" href="supinf_2.html#E8:65"><i><font color="Green">E58</font></i></a>, <a class="ref" href="supinf_2.html#T35" target="_self">Th35</a>, <a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a>, <a class="ref" href="supinf_1.html#D7">SUPINF_1:def 7</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:65"/>
<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> is <a href="supinf_2.html#V1" target="_self">bounded_above</a>
 <b>by </b><i><a class="ref" href="supinf_2.html#D11" target="_self">Def11</a></i>;<br/>


</div>
