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<span class="kw">let </span><font color="Maroon" title="c1">X</font> be   <a href="normsp_1.html#NM1" title="NORMSP_1:NM.1">RealNormSpace</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq1</font>, <font color="Olive" title="b2">seq2</font> being   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">X</font>  st (  for <font color="Olive" title="b3">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b1">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> &amp; <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b1">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b2">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b3">n</font> ) ) &amp; <font color="Olive" title="b2">seq2</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  holds <br/>( <font color="Olive" title="b1">seq1</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b1">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Olive" title="b2">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">seq1</font>, <font color="Maroon" title="c3">seq2</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) ) &amp; <font color="Maroon" title="c3">seq2</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  implies ( <font color="Maroon" title="c2">seq1</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:38"/>
( (  for <font color="Olive" title="b1">n</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  holds <br/> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> &amp; <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Olive" title="b1">n</font> ) ) &amp; <font color="Maroon" title="c3">seq2</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  )
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">seq1</font> is  <a href="lopban_3.html#V2" title="LOPBAN_3:attr.2">norm_summable</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> )</span><br/>

<span class="kw">hence </span><a NAME="E2:38"/>
( <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> is  <a href="series_1.html#V1" title="SERIES_1:attr.1">summable</a>  &amp;  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c2">seq1</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="series_1.html#K3" title="SERIES_1:func.3">Sum</a> <span class="p1"><a href="normsp_0.html#K4" title="NORMSP_0:func.4">||.</a><span class="default"><font color="Maroon" title="c3">seq2</font></span><a href="normsp_0.html#K4" title="NORMSP_0:func.4">.||</a></span> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="series_1.html#T20" title="SERIES_1:th.20">SERIES_1:20</a></span>; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="lopban_3.html#D3" target="_self" title="LOPBAN_3:def.3">LOPBAN_3:def 3</a> <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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