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<b>let </b><font color="Maroon">X</font> be   <a href="csspace.html#NM1">ComplexUnitarySpace</a>;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">Cseq</font> be   <a href="comseq_1.html#NM1">Complex_Sequence</a>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:57"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">Cseq</font> is <a href="comseq_2.html#V2">convergent</a>
 <b>and </b><br/><a NAME="E2:57"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">seq</font> is <a href="clvect_2.html#V1">convergent</a>
 ;<br/>

<a NAME="E3:57"/><i><font color="Green">E48</font></i>: 
<font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font> is <a href="clvect_2.html#V1">convergent</a>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E4:57"/>
<font color="Maroon">Cseq</font> is <a href="comseq_2.html#V1">bounded</a>
 
<b>by </b><i>, </i>;<br/>
<b>then consider </b><font color="Maroon">b</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:57"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>and </b><br/><a NAME="E7:57"/><i><font color="Green">E51</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="comseq_2.html#T8">COMSEQ_2:8</a></i>;<br/>
<a NAME="E8:57"/><i><font color="Green">E60</font></i>: 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="ref" href="csspace.html#T46">CSSPACE:46</a></i>;<br/>
<a NAME="E9:57"/><b>then </b><i><font color="Green">E64</font></i>: 
<font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 0
 
<b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="57_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/><b>assume </b><a NAME="E1:57_1"/>
<font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E2:57_1"/><b>then </b><i><font color="Green">E65</font></i>: 
<font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 <b>by </b><i>, <a class="ref" href="real_2.html#T127">REAL_2:127</a></i>;<br/><b>then consider </b><font color="Maroon">m1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E4:57_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font> holds <br/><span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Olive">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i>, <a class="ref" href="comseq_2.html#D5">COMSEQ_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon">m2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E6:57_1"/><i><font color="Green">E76</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font> holds <br/> <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="clvect_2.html#D2">CLVECT_2:def 2</a></i>;<br/><b>take </b><font color="Maroon">m</font> = <font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font>;<br/><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/><b>assume </b><a NAME="E7:57_1"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m</font>
 ;<br/><a NAME="E8:57_1"/>
<font color="Maroon">m1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m2</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E9:57_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E10:57_1"/><b>then </b><i><font color="Green">E82</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:57_1"/><i><font color="Green">E85</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font>
 <b>by </b><i/>;<br/><a NAME="E12:57_1"/>
<font color="Maroon">m</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/><a NAME="E13:57_1"/><b>then </b>
<font color="Maroon">n</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">m2</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E14:57_1"/><b>then </b>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E15:57_1"/><b>then </b><i><font color="Green">E86</font></i>: 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#D16">CSSPACE:def 16</a></i>;<br/><span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> = 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">X</font>)</span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T10">RLVECT_1:10</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p0">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p0">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p0">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p5">(<span class="default"><span class="p0">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p0">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T43">RLVECT_1:43</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>
;<br/><a NAME="E17:57_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T48">CSSPACE:48</a></i>;<br/><a NAME="E18:57_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T10">CLVECT_1:10</a></i>;<br/><a NAME="E19:57_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i><a class="ref" href="clvect_1.html#T11">CLVECT_1:11</a></i>;<br/><a NAME="E20:57_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span>
 <b>by </b><i><a class="ref" href="csspace.html#T45">CSSPACE:45</a></i>;<br/><a NAME="E21:57_1"/><b>then </b><i><font color="Green">E87</font></i>: 
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="csspace.html#T45">CSSPACE:45</a></i>;<br/><a NAME="E22:57_1"/><i><font color="Green">E88</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="complex1.html#T132">COMPLEX1:132</a></i>;<br/><a NAME="E23:57_1"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="csspace.html#T46">CSSPACE:46</a></i>;<br/><a NAME="E24:57_1"/><b>then </b><i><font color="Green">E89</font></i>: 
<span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , , <a class="ref" href="xreal_1.html#T98">XREAL_1:98</a></i>;<br/><a NAME="E25:57_1"/>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="complex1.html#K17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p3">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span>
 <b>by </b><i>, , <a class="ref" href="xreal_1.html#T66">XREAL_1:66</a></i>;<br/><a NAME="E26:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p4"><a href="csspace.html#K13">||.</a><span class="default"><span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p4"><a href="csspace.html#K13">||.</a><span class="default"><span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/><a NAME="E27:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> <span class="p3">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p4"><a href="csspace.html#K13">||.</a><span class="default"><span class="p5">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E28:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T75">XCMPLX_1:75</a></i>;<br/><a NAME="E29:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K3">+</a> <span class="p2">(<span class="default"><span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="xcmplx_1.html#T63">XCMPLX_1:63</a></i>;<br/><a NAME="E30:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p3"><a href="csspace.html#K13">||.</a><span class="default"><span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K4">*</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K6">/</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="real_1.html#K3">+</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span>
 ;<br/><a NAME="E31:57_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span></span><a href="complex1.html#K17">.|</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span></span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2"><a href="csspace.html#K13">||.</a><span class="default"><span class="p3">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="real_1.html#K4">*</a> <span class="p2"><a href="complex1.html#K17">|.</a><span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="comseq_1.html#K1">.</a> <font color="Maroon">n</font></span>)</span> <a href="complex1.html#K11">-</a> <span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span></span>)</span></span><a href="complex1.html#K17">.|</a></span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xcmplx_1.html#T90">XCMPLX_1:90</a></i>;<br/><a NAME="E32:57_1"/><b>then </b>
<span class="p1"><a href="csspace.html#K13">||.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p4">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span></span>)</span></span><a href="csspace.html#K13">.||</a></span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><b>hence </b><a NAME="E33:57_1"/>
 <a href="csspace.html#K15">dist</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p2">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="ref" href="csspace.html#D16">CSSPACE:def 16</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:57"/>
( <font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font> is <a href="clvect_2.html#V1">convergent</a> &amp;  <a href="clvect_2.html#K1">lim</a> <span class="p1">(<span class="default"><font color="Maroon">Cseq</font> <a href="clvect_3.html#K7" target="_self">*</a> <font color="Maroon">seq</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2">lim</a> <font color="Maroon">Cseq</font></span>)</span> <a href="clvect_1.html#K1">*</a> <span class="p1">(<span class="default"><a href="clvect_2.html#K1">lim</a> <font color="Maroon">seq</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="clvect_2.html#D2">CLVECT_2:def 2</a></i>;<br/>


</div>
