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<span class="kw">let </span><font color="Maroon" title="c1">A</font> be   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="measure5.html#V2" title="MEASURE5:attr.2">closed_interval</a>  <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> holds <br/> ( <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  iff (  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Olive" title="b1">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  ) )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">f</font> be   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">A</font>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  iff (  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  ) )</span><br/>



<a NAME="E1:21"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">A</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c3">f0</font> = <font color="Maroon" title="c2">f</font> as   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>,<a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="relset_1.html#T5" onmouseover="rs('relset_1/T5')" onmouseout="rh()">RELSET_1:5</a></span>;<br/>
<a NAME="E3:21"/>
( <font color="Maroon" title="c3">f0</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  iff (  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c3">f0</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c3">f0</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  ) )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="integr16.html#T11" target="_self" onmouseover="rs('integr16/T11')" onmouseout="rh()">Th11</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:21"/>
( <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  iff (  <a href="comseq_3.html#K5" title="COMSEQ_3:func.5">Re</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  &amp;  <a href="comseq_3.html#K6" title="COMSEQ_3:func.6">Im</a> <font color="Maroon" title="c2">f</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>  ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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