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<b>let </b><font color="Maroon">S</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">T</font> be  non <a href="realset2.html#V3">trivial</a> <a href="normsp_1.html#NM1">RealNormSpace</a>;<br/><b>let </b><font color="Maroon">R</font> be   <a href="ndiff_1.html#NM1">REST</a> of <font color="Maroon">S</font>,<font color="Maroon">T</font>;<br/><b>let </b><font color="Maroon">s</font> be  <a href="ndiff_1.html#V2" target="_self">convergent_to_0</a> <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">S</font>;<br/>







<a NAME="E1:36"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">s</font> <a href="ndiff_1.html#NR1" target="_self">is_not_0</a> 
 
<b>by </b><i/>;<br/>
<a NAME="E2:36"/><i><font color="Green">E45</font></i>: 
( <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <font color="Maroon">s</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font> )
 
<b>by </b><i/>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="36_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:36_1"/>
<font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font>
 <b>by </b><i>, </i>;<br/><a NAME="E2:36_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="normsp_1.html#D2">NORMSP_1:def 2</a></i>;<br/><b>thus </b><span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font> = 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</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">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</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">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</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">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</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">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#D8">SEQ_1:def 8</a></i>
<br/>.= 
<span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="normsp_1.html#D10">NORMSP_1:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p4">(<span class="default"><font color="Maroon">s</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> <a href="real_1.html#K2">"</a> </span>)</span></span>)</span> <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
<br/>.= 
1 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>
<b>by </b><i>, <a class="ref" href="xcmplx_0.html#D7">XCMPLX_0:def 7</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#D9">RLVECT_1:def 9</a></i>
;<br/></div><b>end;</b></div>
<a NAME="E4:36"/><b>then </b><i><font color="Green">E47</font></i>: 
<span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="funct_2.html#R2">=</a> <font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/>
<a NAME="E5:36"/><i><font color="Green">E48</font></i>: 
( <span class="p1">(<span class="default"><span class="p2"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="seq_1.html#K18">"</a> </span>)</span> <a href="ndiff_1.html#K1" target="_self">(#)</a> <span class="p2">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 
<b>by </b><i/>;<br/>
<a NAME="E6:36"/>
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="normsp_1.html#K1">||.</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">S</font></span>)</span></span><a href="normsp_1.html#K1">.||</a></span> )
 
<b>by </b><i>, <a class="ref" href="lopban_1.html#T47">LOPBAN_1:47</a></i>;<br/>
<a NAME="E7:36"/><b>then </b><i><font color="Green">E50</font></i>: 
( <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <span class="p1"><a href="normsp_1.html#K7">||.</a><span class="default"><font color="Maroon">s</font></span><a href="normsp_1.html#K7">.||</a></span> <a href="hidden.html#R1">=</a> 0 )
 
<b>by </b><i><a class="ref" href="normsp_1.html#T5">NORMSP_1:5</a></i>;<br/>
<a NAME="E8:36"/>
( <font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a> 0 <a href="rlvect_1.html#K3">*</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font></span>)</span> )
 
<b>by </b><i>, , , , </i>;<br/>
<b>hence </b><a NAME="E9:36"/>
( <font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font> is <a href="normsp_1.html#V4">convergent</a> &amp;  <a href="normsp_1.html#K8">lim</a> <span class="p1">(<span class="default"><font color="Maroon">R</font> <a href="nfcont_1.html#K3">*</a> <font color="Maroon">s</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">T</font> )
 <b>by </b><i><a class="ref" href="rlvect_1.html#T23">RLVECT_1:23</a></i>;<br/>


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