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<span class="kw">let </span><font color="Maroon" title="c1">X</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">Y</font> being  <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a><br/> for <font color="Olive" title="b2">f</font> being  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b1">Y</font> holds  <span class="p1">(<span class="default"><a href="csspace4.html#K8" title="CSSPACE4:func.8">ComplexBoundedFunctionsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Olive" title="b1">Y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b2">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Olive" title="b2">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">Y</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">f</font> being  <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font> holds  <span class="p1">(<span class="default"><a href="csspace4.html#K8" title="CSSPACE4:func.8">ComplexBoundedFunctionsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Olive" title="b1">f</font></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">f</font> be   <a href="csspace4.html#V1" title="CSSPACE4:attr.1">bounded</a>  <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c1">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c2">Y</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="csspace4.html#K8" title="CSSPACE4:func.8">ComplexBoundedFunctionsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">f</font></span>)</span></span><br/>





<span class="kw">reconsider </span><font color="Maroon" title="c4">f9</font> = <font color="Maroon" title="c3">f</font> as    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  ;<br/>
<a NAME="E2:42"/>
<font color="Maroon" title="c3">f</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="csspace4.html#K4" title="CSSPACE4:func.4">ComplexBoundedFunctions</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#D5" target="_self" title="CSSPACE4:def.5">Def5</a></span>;<br/>
<span class="kw">hence </span><span class="p1">(<span class="default"><a href="csspace4.html#K8" title="CSSPACE4:func.8">ComplexBoundedFunctionsNorm</a> <font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c3">f</font> = 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <span class="p2">(<span class="default"><a href="csspace4.html#K6" title="CSSPACE4:func.6">modetrans</a> <font color="Maroon" title="c4">f9</font>,<font color="Maroon" title="c1">X</font>,<font color="Maroon" title="c2">Y</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#D9" target="_self" title="CSSPACE4:def.9">Def9</a></span>
<br/>.= 
 <a href="measure6.html#NK4" title="MEASURE6:NK.4">sup</a> <span class="p1">(<span class="default"><a href="csspace4.html#K7" title="CSSPACE4:func.7">PreNorms</a> <font color="Maroon" title="c3">f</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="csspace4.html#T17" target="_self" title="CSSPACE4:th.17">Th17</a></span>
;<br/> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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