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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">C</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="partfun1.html#NM1">PartFunc</a> of <font color="Maroon">C</font>, <a href="numbers.html#K2">COMPLEX</a> ;<br/>





<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:71_1"/>
<font color="Maroon">f</font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 ;<br/><a NAME="E2:71_1"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T16" target="_self">Th16</a></i>;<br/><b>then consider </b><font color="Maroon">p</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="E4:71_1"/><i><font color="Green">E35</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="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span> holds <br/> <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Olive">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/><a NAME="E5:71_1"/><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="comseq_1.html#D13">COMSEQ_1:def 13</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c</font> be    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>;<br/><b>assume </b><a NAME="E1:71_1_1"/><i><font color="Green">E40</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span>
 ;<br/><a NAME="E2:71_1_1"/><i><font color="Green">E41</font></i>: 
<font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T17">XBOOLE_1:17</a></i>;<br/><a NAME="E3:71_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E2:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfunct_1.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cfunct_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:71_1_1"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cfunct_1.html#T17" target="_self">Th17</a>, <a class="ref" href="cfunct_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comseq_1.html#D13">COMSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:71_1_1"/>
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E7:71_1"/>
 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/>for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">p</font>
 ;<br/></div>
<b>end;</b></div>

<b>given </b><font color="Maroon">p</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="E2:71"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">c</font> being   <a href="subset_1.html#M1">Element</a> of <font color="Maroon">C</font>  st <font color="Olive">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font></span>)</span> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Olive">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 ;<br/>

<a NAME="E3:71"/><i><font color="Green">E51</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="comseq_1.html#D13">COMSEQ_1:def 13</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="71_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="E1:71_2"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a> <font color="Maroon">Y</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span></span>)</span>
 ;<br/><a NAME="E2:71_2"/><i><font color="Green">E81</font></i>: 
<font color="Maroon">c</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T17" target="_self">Th17</a>, <a class="ref" href="xboole_0.html#D3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:71_2"/><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="txt" href="cfunct_1.html#E2:71"><i><font color="Green">E50</font></i></a>, <a class="txt" href="cfunct_1.html#E3:71"><i><font color="Green">E51</font></i></a>, <a class="ref" href="cfunct_1.html#T17" target="_self">Th17</a></i>;<br/><a NAME="E4:71_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="complex1.html#K18">abs</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">c</font></span>)</span></span><a href="complex1.html#K17">.|</a></span>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T18" target="_self">Th18</a>, <a class="ref" href="comseq_1.html#D13">COMSEQ_1:def 13</a></i>;<br/><b>hence </b><a NAME="E5:71_2"/>
 <a href="complex1.html#K18">abs</a> <span class="p1">(<span class="default"><span class="p2"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T19" target="_self">Th19</a></i>;<br/></div><b>end;</b></div>
<a NAME="E5:71"/><b>then </b>
<span class="p1"><a href="comseq_1.html#K9">|.</a><span class="default"><font color="Maroon">f</font></span><a href="comseq_1.html#K9">.|</a></span> <a href="rfunct_1.html#R3">is_bounded_on</a> <font color="Maroon">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T90">RFUNCT_1:90</a></i>;<br/>
<b>hence </b><a NAME="E6:71"/>
<font color="Maroon">f</font> <a href="cfunct_1.html#R1" target="_self">is_bounded_on</a> <font color="Maroon">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#T16" target="_self">Th16</a></i>;<br/>


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