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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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is  <a href="clvect_1.html#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E1:44"/>
<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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#T25" target="_self" title="CC0SP1:th.25">Th29</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="normsp_0.html#D6" title="NORMSP_0:def.6">NORMSP_0:def 6</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</a> <font color="Maroon" title="c1">X</font> is  <a href="clvect_1.html#V8" title="CLVECT_1:attr.8">ComplexNormSpace-like</a>  )</span><br/>

<a NAME="E2:44"/>
 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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><font color="Olive" title="b3">a</font></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="binop_2.html#K11" title="BINOP_2:func.11">*</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="algstr_0.html#K1" title="ALGSTR_0:func.1">+</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="binop_2.html#K9" title="BINOP_2:func.9">+</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="cc0sp1.html#T25" target="_self" title="CC0SP1:th.25">Th29</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:44"/>
(  <a href="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="cc0sp1.html#K7" title="CC0SP1:func.7">C_Normed_Algebra_of_BoundedFunctions</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="clvect_1.html#D13" title="CLVECT_1:def.13">CLVECT_1:def 13</a>, <a class="ref" href="normsp_0.html#D5" title="NORMSP_0:def.5">NORMSP_0:def 5</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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