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<b>let </b><font color="Maroon">X</font> be    <a href="hidden.html#M1">set</a> ;<br/><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>assume </b><a NAME="E1:91"/><i><font color="Green">E29</font></i>: 
( <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">X</font> &amp; <font color="Maroon">f</font> <a href="rfunct_1.html#R2" target="_self">is_bounded_below_on</a> <font color="Maroon">Y</font> )
 ;<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="E3:91"/><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">X</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">r1</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>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="E5:91"/><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#T42" target="_self">Th42</a>, <a class="ref" href="rfunct_1.html#T21" target="_self">Th21</a></i>;<br/>
<b>take </b><font color="Maroon">r</font> = <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#K5">-</a> <span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r2</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="rfunct_1.html#D10" target="_self">RFUNCT_1:def 10</a><br/>

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

<b>assume </b><a NAME="E6:91"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font></span>)</span> <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="E7:91"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> <a href="xboole_0.html#K2">\/</a> <font color="Maroon">Y</font> &amp; <font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="91_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:91_1_1"/>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font> or <font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> )
 </span><b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_0.html#D2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_1"><b>suppose </b></a><a NAME="E1:91_1_1_1"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</font>
 ;<br/><div class="add"><a NAME="E2:91_1_1_1"/><b>then </b>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">X</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>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:91_1_1_1"/><b>then </b><i><font color="Green">E33</font></i>: 
<font color="Maroon">r1</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#T43" target="_self">Th43</a></i>;<br/><a NAME="E4:91_1_1_1"/>
 <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> <font color="Maroon">r1</font>
 <b>by </b><i><a class="ref" href="absvalue.html#T11">ABSVALUE:11</a></i>;<br/><a NAME="E5:91_1_1_1"/><b>then </b><i><font color="Green">E63</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> <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#T46" target="_self">Th46</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:91_1_1_1"/>
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="E7:91_1_1_1"/><b>then </b>
<font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</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="real_1.html#K5">-</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T47" target="_self">Th47</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E8:91_1_1_1"/>
<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>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="91_1_1_2"><b>suppose </b></a><a NAME="E1:91_1_1_2"/>
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font>
 ;<br/><div class="add"><a NAME="E2:91_1_1_2"/><b>then </b>
<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>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T45" target="_self">Th45</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:91_1_1_2"/><b>then </b><i><font color="Green">E64</font></i>: 
<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#T44" target="_self">Th44</a></i>;<br/><a NAME="E4:91_1_1_2"/>
 <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="E5:91_1_1_2"/><b>then </b><i><font color="Green">E69</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">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#T46" target="_self">Th46</a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E6:91_1_1_2"/>
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="E7:91_1_1_2"/><b>then </b>
<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="real_1.html#K5">-</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> <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#K5">-</a> 0
 <b>by </b><i><a class="ref" href="rfunct_1.html#T47" target="_self">Th47</a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/><b>hence </b><a NAME="E8:91_1_1_2"/>
<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>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E9:91"/>
<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>
 ;<br/>


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