<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>let </b><font color="Maroon">c<sub>2</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#K1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:40"/><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>2</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<b>set </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>
<div><i><font color="Green">E27</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="40_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:40_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K6">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><b>then consider </b><font color="Maroon">c<sub>5</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:40_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font> )
 ;<br/><a NAME="E4:40_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:40"><i><font color="Green">E26</font></i></a>, <a class="txt" href="pscomp_1.html#E3:40_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="seq_4.html#D5">SEQ_4:def 5</a></i>;<br/><b>hence </b><a NAME="E5:40_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:40_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:40_2"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K6">+</a> <font color="Maroon">c<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="40_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>5</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>assume </b><a NAME="E1:40_2_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E2:40_2_1"/><b>then </b>
<font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K6">+</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><a NAME="E3:40_2_1"/><b>then </b>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:40_2"><i><font color="Green">E28</font></i></a></i>;<br/><b>hence </b><a NAME="E4:40_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>5</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div><a NAME="E3:40_2"/><b>then </b>
 <a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>4</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="seq_4.html#T60">SEQ_4:60</a></i>;<br/><b>hence </b><a NAME="E4:40_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T22">XREAL_1:22</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:40"/>
 <a href="pscomp_1.html#K3" target="_self">inf</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="measure6.html#K6">+</a> <font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:40"><i><font color="Green">E27</font></i></a>, <a class="ref" href="seq_4.html#T61">SEQ_4:61</a></i>;<br/>


</div>
