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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><a NAME="E1:16"/><i><font color="Green">E11</font></i>: 
(  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> &amp;  <a href="supinf_2.html#K1">0.</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="supinf_2.html#K2">+</a> <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> )
 ;<br/>

<a NAME="E2:16"/><i><font color="Green">E12</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is   <a href="real_1.html#NM1">Real</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green">E13</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is  not   <a href="real_1.html#NM1">Real</a>
 ;<br/>

<a NAME="E2:16_1"/><i><font color="Green">E14</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="supinf_1.html#K3">REAL</a>  or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K7">{</a><span class="default"><a href="supinf_1.html#K2">-infty</a> ,<a href="supinf_1.html#K1">+infty</a> </span><a href="domain_1.html#K7">}</a></span> )
 
<b>by </b><i><a class="ref" href="xxreal_0.html#D4">XXREAL_0:def 4</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/>
<a NAME="E3:16_1"/>
<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> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a> 
 ;<br/>

<a NAME="E2:16_1_1"/><b>then </b>
 <a href="supinf_1.html#K1">+infty</a>  <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="measure6.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="supinf_2.html#D2">SUPINF_2:def 2</a></i>;<br/>
<b>hence </b><a NAME="E3:16_1_1"/>
contradiction
 <b>by </b><i><a class="ref" href="supinf_1.html#T24">SUPINF_1:24</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:16_1"/>
contradiction
 <b>by </b><i><a class="txt" href="measure6.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure6.html#E1:16_1"><i><font color="Green">E13</font></i></a>, <a class="txt" href="measure6.html#E2:16_1"><i><font color="Green">E14</font></i></a>, <a class="ref" href="supinf_2.html#T19">SUPINF_2:19</a>, <a class="ref" href="tarski.html#D2">TARSKI:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:16"/><b>then </b><i><font color="Green">E13</font></i>: 
( not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  &amp; not <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  )
 
;<br/>
<a NAME="E4:16"/><i><font color="Green">E14</font></i>: 
( not  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K1">+infty</a>  &amp; not  <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a>  <a href="supinf_1.html#K2">-infty</a>  )
 
<b>by </b><i><a class="ref" href="supinf_2.html#D1">SUPINF_2:def 1</a></i>;<br/>
<a NAME="E5:16"/>
( <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>1</sub></font></span>)</span> <a href="supinf_2.html#K4">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="supinf_2.html#K4">-</a> <a href="supinf_2.html#K1">0.</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="measure6.html#E2:16"><i><font color="Green">E12</font></i></a>, <a class="ref" href="measure6.html#T8" target="_self">Th8</a>, <a class="ref" href="measure6.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="measure6.html#E1:16"><i><font color="Green">E11</font></i></a>, <a class="txt" href="measure6.html#E3:16"><i><font color="Green">E13</font></i></a>, <a class="txt" href="measure6.html#E4:16"><i><font color="Green">E14</font></i></a>, <a class="ref" href="supinf_2.html#T15">SUPINF_2:15</a></i>;<br/>


</div>
