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<b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>

<b>set </b><font color="Maroon">X</font> = <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span>;<br/>
<a NAME="E1:16"/><i><font color="Green">E45</font></i>: 
<span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> is <a href="seq_4.html#V3" target="_self">bounded</a>
 
<b>by </b><i/>;<br/>
<a NAME="E2:16"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> is <a href="seq_4.html#V2" target="_self">bounded_below</a>
 
<b>by </b><i/>;<br/>
<a NAME="E3:16"/><i><font color="Green">E48</font></i>: 
<span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> is <a href="seq_4.html#V1" target="_self">bounded_above</a>
 
<b>by </b><i>, </i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:16_1"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> holds <br/><font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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="E1:16_1_1"/><i><font color="Green">E49</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><b>take </b><font color="Maroon">r1</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E2:16_1_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_1_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E4:16_1_1"/>
<font color="Maroon">r1</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">s</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:16_1"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> &amp; <font color="Olive">p</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <font color="Olive">s</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:16"/>
 <a href="seq_4.html#K5" target="_self">lower_bound</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2"><b>now </b></a><div class="add"><b>thus </b><a NAME="E1:16_2"/>
for <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> holds <br/><font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_2_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">s</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="E1:16_2_1"/><i><font color="Green">E50</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">s</font>
 ;<br/><b>take </b><font color="Maroon">r1</font> = <font color="Maroon">r</font>;<br/><b>thus </b><a NAME="E2:16_2_1"/>
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E3:16_2_1"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="real_1.html#K1">-</a> 0
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:16_2_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="xcmplx_0.html#K4">-</a> <font color="Maroon">s</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="xcmplx_0.html#K2">+</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><b>hence </b><a NAME="E5:16_2_1"/>
<font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r1</font>
 ;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:16_2"/>
for <font color="Olive">s</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>   st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">s</font> holds <br/> ex <font color="Olive">p</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> &amp; <font color="Maroon">r</font> <a href="xcmplx_0.html#K6">-</a> <font color="Olive">s</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">p</font> )
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E7:16"/>
 <a href="seq_4.html#K4" target="_self">upper_bound</a> <span class="p1"><a href="seq_4.html#K1" target="_self">{</a><span class="default"><font color="Maroon">r</font></span><a href="seq_4.html#K1" target="_self">}</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, </i>;<br/>


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