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<b>let </b><font color="Maroon">c<sub>1</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="E1:27"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V2">bounded_below</a>
 ;<br/>

<a NAME="E2:27"/><b>then </b><i><font color="Green">E18</font></i>: 
 <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font> is <a href="seq_4.html#V1">bounded_above</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E23"><i><font color="Green">Lemma15</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>2</sub></font> =  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>;<br/>
<div><i><font color="Green">E19</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="27_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</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="E1:27_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:27_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E3:27_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E2:27"><i><font color="Green">E18</font></i></a>, <a class="ref" href="seq_4.html#D4">SEQ_4:def 4</a></i>;<br/><b>hence </b><a NAME="E4:27_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>3</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="E1:27_2"/><i><font color="Green">E20</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> 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>3</sub></font>
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="27_2_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="E1:27_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><a NAME="E2:27_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="pscomp_1.html#K4" target="_self">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span>
 ;<br/><a NAME="E3:27_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E1:27_2"><i><font color="Green">E20</font></i></a></i>;<br/><a NAME="E4:27_2_1"/><b>then </b>
 <a href="xcmplx_0.html#K4">-</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><b>hence </b><a NAME="E5:27_2_1"/>
<font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/></div><b>end;</b></div><a NAME="E3:27_2"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p3">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="seq_4.html#T62">SEQ_4:62</a></i>;<br/><b>hence </b><a NAME="E4:27_2"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:27"/>
 <a href="pscomp_1.html#K3" target="_self">inf</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="pscomp_1.html#K2" target="_self">sup</a> <span class="p2">(<span class="default"><a href="pscomp_1.html#K4" target="_self">-</a> <font color="Maroon">c<sub>1</sub></font></span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E3:27"><i><font color="Green">E19</font></i></a>, <a class="ref" href="seq_4.html#T61">SEQ_4:61</a></i>;<br/>


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