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<span class="kw">set </span><font color="Maroon" title="c2">C</font> =  <a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font>;<br/>
<a NAME="E1:50_1_1"/><span class="lab"><font color="Green" title="E30">A1</font></span>: 
<span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p3">(<span class="default"><a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cc0sp2.html#T25" target="_self" title="CC0SP2:th.25">Th27a</a></span>;<br/>
<a NAME="E2:50_1_1"/><span class="lab"><font color="Green" title="E31">A2</font></span>: 
 for <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span><br/>  for <font color="Olive" title="b3">a</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a> holds <br/> ( ( <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span> ) &amp; ( <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <span class="p1">(<span class="default"><a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font></span>)</span> implies <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  ) &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b3">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b1">x</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> &amp; <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">x</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b2">y</font></span>)</span></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b1">x</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1"><a href="normsp_0.html#K1" title="NORMSP_0:func.1">||.</a><span class="default"><font color="Olive" title="b2">y</font></span><a href="normsp_0.html#K1" title="NORMSP_0:func.1">.||</a></span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cc0sp2.html#T25" target="_self" title="CC0SP2:th.25">Th27a</a>, <a class="ref" href="cc0sp2.html#T26" target="_self" title="CC0SP2:th.26">Th27b</a>, <a class="ref" href="cc0sp2.html#T27" target="_self" title="CC0SP2:th.27">Th27c</a>, <a class="ref" href="cc0sp2.html#T28" target="_self" title="CC0SP2:th.28">Th27d</a></span>;<br/>
<span class="kw">thus </span><a NAME="E3:50_1_1"/>
(  <a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is  <a href="normsp_0.html#V3" title="NORMSP_0:attr.3">discerning</a>  &amp;  <a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is  <a href="normsp_0.html#V4" title="NORMSP_0:attr.4">reflexive</a>  &amp;  <a href="cc0sp2.html#K5" title="CC0SP2:func.5">C_Normed_Algebra_of_ContinuousFunctions</a> <font color="Maroon" title="c1">X</font> is  <a href="clvect_1.html#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</a>, <a class="ref" href="normsp_0.html#D6" title="NORMSP_0:def.6">NORMSP_0:def 6</a>, <a class="ref" href="clvect_1.html#D13" title="CLVECT_1:def.13">CLVECT_1:def 13</a>, <a class="txt" href="#E1:50_1_1"><span class="lab"><font color="Green" title="E30">A1</font></span></a>, <a class="txt" href="#E2:50_1_1"><span class="lab"><font color="Green" title="E31">A2</font></span></a></span>; <a class="txt" 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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