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<b>let </b><font color="Maroon" title="c1">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">C</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <br/> for <font color="Olive" title="b2">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Olive" title="b1">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b3">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b2">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font> holds <br/><font color="Olive" title="b3">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b2">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font></span><br/><b>let </b><font color="Maroon" title="c2">C</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> <br/> for <font color="Olive" title="b2">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Olive" title="b1">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font> holds <br/><font color="Olive" title="b2">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Olive" title="b1">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font></span><br/><b>let </b><font color="Maroon" title="c3">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of <font color="Maroon" title="c2">C</font>, <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">r</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a>   st <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font> holds <br/><font color="Olive" title="b1">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font></span><br/><b>let </b><font color="Maroon" title="c4">r</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K2" title="NUMBERS:func.2">COMPLEX</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font> implies <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font> )</span><br/>







<b>assume </b><a NAME="E1:70"/>
<font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c4">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font></span><br/>

<a NAME="E2:70"/><b>then </b>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#D3" target="_self" title="CFUNCT_1:def.3">Def3</a></i>;<br/>
<a NAME="E3:70"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Maroon" title="c4">r</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="valued_1.html#K26" title="VALUED_1:func.26">(#)</a> <span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><font color="Maroon" title="c3">f</font></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="rfunct_1.html#T97" title="RFUNCT_1:th.97">RFUNCT_1:97</a></i>;<br/>
<a NAME="E4:70"/><b>then </b>
<span class="p1"><a href="valued_1.html#K55" title="VALUED_1:func.55">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span><a href="valued_1.html#K55" title="VALUED_1:func.55">.|</a></span> <a href="rfunct_1.html#R3" title="RFUNCT_1:pred.3">is_bounded_on</a> <font color="Maroon" title="c1">Y</font>
 
<b>by </b><i><a class="ref" href="cfunct_1.html#T39" target="_self" title="CFUNCT_1:th.39">Th39</a></i>;<br/>
<b>hence </b><a NAME="E5:70"/>
<font color="Maroon" title="c4">r</font> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c3">f</font> <a href="cfunct_1.html#R1" title="CFUNCT_1:pred.1">is_bounded_on</a> <font color="Maroon" title="c1">Y</font>
 <b>by </b><i><a class="ref" href="cfunct_1.html#D3" target="_self" title="CFUNCT_1:def.3">Def3</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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