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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="clvect_1.html#NM5" title="CLVECT_1:NM.5">ComplexNormSpace</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font><br/>  for <font color="Olive" title="b2">rseq1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; (  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b2">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b3">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b3">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Olive" title="b1">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b3">n</font></span>)</span></span>)</span> ) holds <br/>( <font color="Olive" title="b1">seq</font> is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  iff <font color="Olive" title="b2">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq</font> be   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">rseq1</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; (  for <font color="Olive" title="b2">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Olive" title="b1">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b2">n</font></span>)</span></span>)</span> ) holds <br/>( <font color="Maroon" title="c2">seq</font> is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  iff <font color="Olive" title="b1">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">rseq1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) implies ( <font color="Maroon" title="c2">seq</font> is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  ) )</span><br/>





<span class="kw">assume </span><a NAME="E1:40"/>
( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; (  for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds  <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> ) )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq</font> is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )</span><br/>

<a NAME="E2:40"/><span class="kw">then </span>
 for <font color="Olive" title="b1">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="valued_0.html#V10" title="VALUED_0:attr.10">non-increasing</a>  &amp; <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  &amp; <font color="Maroon" title="c3">rseq1</font> <a href="nat_1.html#K8" title="NAT_1:func.8">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p2">(<span class="default">2 <a href="power.html#K2" title="POWER:func.2">to_power</a> <font color="Olive" title="b1">n</font></span>)</span></span>)</span> )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="clopban3.html#T2" target="_self" onmouseover="rs('clopban3/T2')" onmouseout="rh()">Th2</a></span>;<br/>
<a NAME="E3:40"/><span class="kw">then </span>
( <span class="p1"><a href="normsp_0.html#K5" title="NORMSP_0:func.5">||.</a><span class="default"><font color="Maroon" title="c2">seq</font></span><a href="normsp_0.html#K5" title="NORMSP_0:func.5">.||</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T31" onmouseover="rs('series_1/T31')" onmouseout="rh()">SERIES_1:31</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:40"/>
( <font color="Maroon" title="c2">seq</font> is  <a href="clopban3.html#V2" title="CLOPBAN3:attr.2">norm_summable</a>  iff <font color="Maroon" title="c3">rseq1</font> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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