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<span class="kw">let </span><font color="Maroon" title="c1">S</font>, <font color="Maroon" title="c2">T</font> be  non  <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a>  <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="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">R</font> being  <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font><br/> for <font color="Olive" title="b2">s</font> being  <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">convergent_to_0</a>  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Olive" title="b1">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">s</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b2">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c3">R</font> be   <a href="ndiff_1.html#NM1" title="NDIFF_1:NM.1">REST</a> of <font color="Maroon" title="c1">S</font>,<font color="Maroon" title="c2">T</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">s</font> being  <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">convergent_to_0</a>  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font> holds <br/> ( <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">s</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Olive" title="b1">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c4">s</font> be   <a href="ndiff_1.html#V1" title="NDIFF_1:attr.1">convergent_to_0</a>  <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">S</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font> )</span><br/>







<a NAME="E1:35"/><span class="lab"><font color="Green" title="E23">A1</font></span>: 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></span>;<br/>
<a NAME="E2:35"/><span class="lab"><font color="Green" title="E24">A2</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="struct_0.html#V12" title="STRUCT_0:attr.12">non-zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D4" target="_self" title="NDIFF_1:def.4">Def4</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c5">n</font> be    <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> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span><br/><a NAME="E1:35_1"/>
<font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">S</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35"><span class="lab"><font color="Green" title="E24">A2</font></span></a>, <a class="ref" href="ndiff_1.html#T7" target="_self" title="NDIFF_1:th.7">Th7</a></span>;<br/><a NAME="E2:35_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A3</font></span>: 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D2" title="NORMSP_1:def.2">NORMSP_1:def 2</a></span>;<br/><span class="kw">thus </span><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D2" target="_self" title="NDIFF_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D2" target="_self" title="NDIFF_1:def.2">Def2</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="seq_1.html#K1" title="SEQ_1:func.1">.</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="valued_1.html#T10" title="VALUED_1:th.10">VALUED_1:10</a></span>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="normsp_1.html#D10" title="NORMSP_1:def.10">NORMSP_1:def 10</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1" title="NORMSP_1:func.1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D10" title="RLVECT_1:def.10">RLVECT_1:def 10</a>, <a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
<br/>.= 
1 <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:35_1"><span class="lab"><font color="Green" title="E25">A3</font></span></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c5">n</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D11" title="RLVECT_1:def.11">RLVECT_1:def 11</a></span>
; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A4</font></span>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#T113" title="FUNCT_2:th.113">FUNCT_2:113</a></span>;<br/>
<a NAME="E5:35"/><span class="lab"><font color="Green" title="E26">A5</font></span>: 
<font color="Maroon" title="c4">s</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D4" target="_self" title="NDIFF_1:def.4">Def4</a></span>;<br/>
<a NAME="E6:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E27">A6</font></span>: 
<span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="lopban_1.html#T47" title="LOPBAN_1:th.47">LOPBAN_1:47</a></span>;<br/>
<a NAME="E7:35"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <font color="Maroon" title="c4">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">S</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D4" target="_self" title="NDIFF_1:def.4">Def4</a></span>;<br/>
<a NAME="E8:35"/><span class="kw">then </span>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="normsp_1.html#K1" title="NORMSP_1: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> <font color="Maroon" title="c1">S</font></span>)</span></span><a href="normsp_1.html#K1" title="NORMSP_1:func.1">.||</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="ref" href="lopban_1.html#T47" title="LOPBAN_1:th.47">LOPBAN_1:47</a></span>;<br/>
<a NAME="E9:35"/><span class="kw">then </span><span class="lab"><font color="Green" title="E28">A7</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</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="normsp_1.html#T5" title="NORMSP_1:th.5">NORMSP_1:5</a></span>;<br/>
<a NAME="E10:35"/>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7" title="NORMSP_1:func.7">||.</a><span class="default"><font color="Maroon" title="c4">s</font></span><a href="normsp_1.html#K7" title="NORMSP_1:func.7">.||</a></span> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span> <a href="ndiff_1.html#K1" title="NDIFF_1:func.1">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="ndiff_1.html#D5" target="_self" title="NDIFF_1:def.5">Def5</a></span>;<br/>
<a NAME="E11:35"/><span class="kw">then </span>
 <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">*</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:35"><span class="lab"><font color="Green" title="E26">A5</font></span></a>, <a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E9:35"><span class="lab"><font color="Green" title="E28">A7</font></span></a>, <a class="ref" href="ndiff_1.html#T14" target="_self" title="NDIFF_1:th.14">Th14</a>, <a class="ref" href="lopban_1.html#T47" title="LOPBAN_1:th.47">LOPBAN_1:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E12:35"/>
( <font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font> is  <a href="normsp_1.html#V3" title="NORMSP_1:attr.3">convergent</a>  &amp;  <a href="normsp_1.html#K8" title="NORMSP_1:func.8">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">R</font> <a href="funct_2.html#K8" title="FUNCT_2:func.8">/*</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c2">T</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:35"><span class="lab"><font color="Green" title="E25">A4</font></span></a>, <a class="txt" href="#E1:35"><span class="lab"><font color="Green" title="E23">A1</font></span></a>, <a class="txt" href="#E6:35"><span class="lab"><font color="Green" title="E27">A6</font></span></a>, <a class="ref" href="ndiff_1.html#T13" target="_self" title="NDIFF_1:th.13">Th13</a>, <a class="ref" href="rlvect_1.html#T23" title="RLVECT_1:th.23">RLVECT_1:23</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
