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

<b>let </b><font color="Maroon">Y</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be  <a href="seq_1.html#V1">real-yielding</a> <a href="funct_1.html#NM1">Function</a>;<br/>



<b>thus </b><a NAME="E1:78"/>
( <font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">Y</font> implies  ex <font color="Olive">r</font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <br/>for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">r</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/><i><font color="Green">E29</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 ;<br/>

<a NAME="E2:78_1"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">r1</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E4:78_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T17" target="_self">Th17</a></i>;<br/>
<a NAME="E5:78_1"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a>, <a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>
<b>then consider </b><font color="Maroon">r2</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><br/><a NAME="E7:78_1"/><i><font color="Green">E31</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>take </b><font color="Maroon">g</font> = <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span>;<br/>

<b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E8:78_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/>

<a NAME="E9:78_1"/><i><font color="Green">E33</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E10:78_1"/><i><font color="Green">E63</font></i>: 
0 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/>
<a NAME="E11:78_1"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E12:78_1"/>
<font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rfunct_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E13:78_1"/><b>then </b>
<font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T39" target="_self">Th39</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E14:78_1"/><b>then </b><i><font color="Green">E69</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="real_1.html#K3">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T38" target="_self">Th38</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E15:78_1"/><i><font color="Green">E70</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r2</font>
 
<b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/>
<a NAME="E16:78_1"/><i><font color="Green">E71</font></i>: 
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> 0
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T36" target="_self">Th36</a>, <a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/>
<a NAME="E17:78_1"/>
<font color="Maroon">r2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a>, <a class="ref" href="rfunct_1.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E18:78_1"/><b>then </b>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T42" target="_self">Th42</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:78_1"/><b>then </b><i><font color="Green">E72</font></i>: 
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T43" target="_self">Th43</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/>
<a NAME="E20:78_1"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r1</font></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> <span class="p2">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="real_1.html#K1">-</a> <font color="Maroon">g</font>
 
;<br/>
<b>hence </b><a NAME="E21:78_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">g</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T40" target="_self">Th40</a>, <a class="ref" href="rfunct_1.html#T44" target="_self">Th44</a>, <a class="ref" href="absvalue.html#T12">ABSVALUE:12</a></i>;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon">r</font> being  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <b> such that </b><a NAME="E2:78"/><i><font color="Green">E73</font></i>: 
for <font color="Olive">c</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Olive">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:78_2"/><b>then </b><i><font color="Green">E75</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E3:78_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><b>hence </b><a NAME="E4:78_2"/>
<font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:78"/><b>then </b><i><font color="Green">E76</font></i>: 
<font color="Maroon">f</font> <a href="rfunct_1.html#R1" target="_self">is_bounded_above_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T17" target="_self">Th17</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:78_3"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="xboole_0.html#K3">/\</a> <span class="p1">(<span class="default"><a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:78_3"/><b>then </b>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T25" target="_self">Th25</a></i>;<br/><a NAME="E3:78_3"/><b>then </b><i><font color="Green">E77</font></i>: 
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T26">XREAL_1:26</a></i>;<br/><a NAME="E4:78_3"/>
 <a href="real_1.html#K1">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><b>hence </b><a NAME="E5:78_3"/>
 <a href="xcmplx_0.html#K4">-</a> <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">f</font> <a href="seq_1.html#K1">.</a> <font color="Maroon">c</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:78"/><b>then </b>
<font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>hence </b><a NAME="E7:78"/>
<font color="Maroon">f</font> <a href="rfunct_1.html#R3" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T26" target="_self">Th26</a>, <a class="ref" href="rfunct_1.html#T22" target="_self">Th22</a></i>;<br/>


</div>
