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<span class="kw">let </span><font color="Maroon" title="c1">seq</font>, <font color="Maroon" title="c2">cseq</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">cseq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> implies ( <font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  iff <font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:7"/><span class="lab"><font color="Green" title="E8">A1</font></span>: 
<font color="Maroon" title="c2">cseq</font> <a href="relset_1.html#R2" title="RELSET_1:pred.2">=</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  iff <font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  )</span><br/>

<span class="kw">thus </span><a NAME="E2:7"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  implies <font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  )
  <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  implies <font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="7_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:7_1"/><span class="lab"><font color="Green" title="E9">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> </span><br/>

<a NAME="E2:7_1"/><span class="kw">then </span>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <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="cfdiff_1.html#D1" title="CFDIFF_1:def.1">CFDIFF_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E10">A3</font></span>:  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span> = 
<a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="comseq_2.html#T18" title="COMSEQ_2:th.18">COMSEQ_2:18</a></span>
<br/>.= 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 

;<br/>
<a NAME="E4:7_1"/>
( <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  &amp; <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7_1"><span class="lab"><font color="Green" title="E9">A2</font></span></a>, <a class="ref" href="comseq_1.html#T38" title="COMSEQ_1:th.38">COMSEQ_1:38</a>, <a class="ref" href="comseq_2.html#T17" title="COMSEQ_2:th.17">COMSEQ_2:17</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:7_1"/>
<font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:7_1"><span class="lab"><font color="Green" title="E10">A3</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" title="CFDIFF_1:def.1">CFDIFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>

<span class="kw">assume </span><a NAME="E3:7"/><span class="lab"><font color="Green" title="E9">A4</font></span>: 
<font color="Maroon" title="c2">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> </span><br/>

<a NAME="E4:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A5</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="cfdiff_2.html#E6"><span class="lab"><font color="Green" title="E6">Lm5</font></span></a></span>;<br/>
<a NAME="E5:7"/>
<span class="p1">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</span> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="comseq_2.html#T17" title="COMSEQ_2:th.17">COMSEQ_2:17</a></span>;<br/>
<a NAME="E6:7"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="complex1.html#K10" title="COMPLEX1:func.10">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T22" title="CFUNCT_1:th.22">CFUNCT_1:22</a></span>;<br/>
<a NAME="E7:7"/><span class="kw">then </span><span class="lab"><font color="Green" title="E11">A6</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T26" title="CFUNCT_1:th.26">CFUNCT_1:26</a></span>;<br/>
<a NAME="E8:7"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="valued_1.html#K25" title="VALUED_1:func.25">(#)</a> <font color="Maroon" title="c1">seq</font></span>)</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="txt" href="#E1:7"><span class="lab"><font color="Green" title="E8">A1</font></span></a>, <a class="txt" href="#E3:7"><span class="lab"><font color="Green" title="E9">A4</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" title="CFDIFF_1:def.1">CFDIFF_1:def 1</a></span>;<br/>
<a NAME="E9:7"/><span class="kw">then </span>
<a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font></span>)</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="txt" href="#E7:7"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="ref" href="comseq_2.html#T18" title="COMSEQ_2:th.18">COMSEQ_2:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:7"/>
<font color="Maroon" title="c1">seq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:7"><span class="lab"><font color="Green" title="E11">A6</font></span></a>, <a class="txt" href="#E4:7"><span class="lab"><font color="Green" title="E10">A5</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" title="CFDIFF_1:def.1">CFDIFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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