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<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="E1:87"/><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/>

<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="E3:87"/><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/> <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">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T24" target="_self">Th24</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="87_1"><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="E1:87_1"/>
<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> <span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:87_1"/><b>then </b><i><font color="Green">E31</font></i>: 
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</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> <span class="p1">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> )
 <b>by </b><i><a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:87_1"/><b>then </b>
( <font color="Maroon">c</font> <a href="hidden.html#R2">in</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="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/><a NAME="E4:87_1"/><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="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E5:87_1"/><b>then </b>
 <a href="complex1.html#K18">abs</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">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T38" target="_self">Th38</a></i>;<br/><b>hence </b><a NAME="E6:87_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K20">abs</a> <font color="Maroon">f</font></span>)</span> <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">r1</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T39" target="_self">Th39</a>, <a class="ref" href="seq_1.html#D10">SEQ_1:def 10</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:87"/>
 <a href="seq_1.html#K20">abs</a> <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#T24" target="_self">Th24</a></i>;<br/>

<a NAME="E6:87"/>
<span class="p1">(<span class="default"><a href="real_1.html#K1">-</a> 1</span>)</span> <a href="seq_1.html#K12">(#)</a> <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#T36" target="_self">Th36</a>, <a class="ref" href="rfunct_1.html#T29" target="_self">Th29</a></i>;<br/>
<b>hence </b><a NAME="E7:87"/>
 <a href="seq_1.html#K15">-</a> <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/>;<br/>


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