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<b>let </b><font color="Maroon">c<sub>1</sub></font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="hidden.html#M1">set</a> ;<br/><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="clvect_1.html#NM5">ComplexNormSpace</a>;<br/>



<a NAME="E1:50"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="cfunct_1.html#NM1">Complex</a> holds <br/>( ( <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0 implies <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> ) &amp; ( <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> implies <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> 0 ) &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>3</sub></font> <a href="clvect_1.html#K1">*</a> <font color="Olive">b<sub>1</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><font color="Olive">b<sub>3</sub></font></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> &amp; <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Olive">b<sub>1</sub></font> <a href="rlvect_1.html#K2">+</a> <font color="Olive">b<sub>2</sub></font></span>)</span></span><a href="clvect_1.html#K5">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="clvect_1.html#K5">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="clvect_1.html#K5">||.</a><span class="default"><font color="Olive">b<sub>2</sub></font></span><a href="clvect_1.html#K5">.||</a></span> )
 
<b>by </b><i><a class="ref" href="csspace4.html#T25" target="_self">Th25</a></i>;<br/>
<b>hence </b><a NAME="E2:50"/>
 <a href="csspace4.html#K9" target="_self">C_NormSpace_of_BoundedFunctions</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> is <a href="clvect_1.html#V5">ComplexNormSpace-like</a>
 <b>by </b><i><a class="ref" href="clvect_1.html#D10">CLVECT_1:def 10</a></i>;<br/>


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