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<b>let </b><font color="Maroon" title="c1">M</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> ;<br/><b>let </b><font color="Maroon" title="c2">V</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>;<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="c1">M</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">V</font>;<br/><b>let </b><font color="Maroon" title="c4">Y</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>







<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="48_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c5">p</font> = 0;<br/><b>let </b><font color="Maroon" title="c6">c</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c1">M</font>;<br/><b>assume </b><a NAME="E1:48_1"/>
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c4">Y</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span></span>)</span>
 ;<br/><a NAME="E2:48_1"/><b>then </b><i><font color="Green" title="E20">A1</font></i>: 
<font color="Maroon" title="c6">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><a NAME="E3:48_1"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T130" title="COMPLEX1:th.130">COMPLEX1:130</a></i>;<br/><a NAME="E4:48_1"/><b>then </b>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">f</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 0
 <b>by </b><i><a class="ref" href="clvect_1.html#D10" title="CLVECT_1:def.10">CLVECT_1:def 10</a></i>;<br/><b>hence </b><a NAME="E5:48_1"/>
<span class="p1"><a href="clvect_1.html#K5" title="CLVECT_1:func.5">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <font color="Maroon" title="c6">c</font></span>)</span></span><a href="clvect_1.html#K5" title="CLVECT_1:func.5">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">p</font>
 <b>by </b><i><a class="txt" href="vfunct_2.html#E2:48_1"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="vfunct_2.html#D4" target="_self" title="VFUNCT_2:def.4">Def4</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:48"/>
<a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  <a href="vfunct_2.html#K4" title="VFUNCT_2:func.4">(#)</a> <font color="Maroon" title="c3">f</font> <a href="vfunct_2.html#R1" title="VFUNCT_2:pred.1">is_bounded_on</a> <font color="Maroon" title="c4">Y</font>
 <b>by </b><i><a class="ref" href="vfunct_2.html#D7" target="_self" title="VFUNCT_2:def.7">Def7</a></i>;<br/>


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