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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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  iff ( <font color="Maroon" title="c2">cseq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  ) ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:6"/><span class="lab"><font color="Green" title="E7">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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  iff ( <font color="Maroon" title="c2">cseq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  ) )</span><br/>

<span class="kw">thus </span><a NAME="E2:6"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  implies ( <font color="Maroon" title="c2">cseq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  ) )
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  implies ( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  ) )</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="6_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:6_1"/><span class="lab"><font color="Green" title="E8">A2</font></span>: 
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )</span><br/>

<a NAME="E2:6_1"/><span class="kw">then </span>
 <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfdiff_1.html#D1" onmouseover="rs('cfdiff_1/D1')" onmouseout="rh()">CFDIFF_1:def 1</a></span>;<br/>
<span class="kw">then </span><span class="lab"><font color="Green" title="E9">A3</font></span>:  <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">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#K5" title="NUMBERS:func.5">0</a>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="comseq_2.html#T18" onmouseover="rs('comseq_2/T18')" onmouseout="rh()">COMSEQ_2:18</a></span>
<br/>.= 
 <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 

;<br/>
<a NAME="E4:6_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="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</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#V2" title="COMSEQ_2:attr.2">convergent</a>  )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6_1"><span class="lab"><font color="Green" title="E8">A2</font></span></a>, <a class="ref" href="comseq_1.html#T38" onmouseover="rs('comseq_1/T38')" onmouseout="rh()">COMSEQ_1:38</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:6_1"/>
( <font color="Maroon" title="c2">cseq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:6_1"><span class="lab"><font color="Green" title="E9">A3</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" onmouseover="rs('cfdiff_1/D1')" onmouseout="rh()">CFDIFF_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="E3:6"/><span class="lab"><font color="Green" title="E8">A4</font></span>: 
( <font color="Maroon" title="c2">cseq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c2">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )</span><br/>

<a NAME="E4:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="cfdiff_2.html#E5"><span class="lab"><font color="Green" title="E5">Lm4</font></span></a></span>;<br/>
<a NAME="E5:6"/>
<span class="p1">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</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#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a></span>;<br/>
<a NAME="E6:6"/><span class="kw">then </span>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="xcmplx_0.html#K4" title="XCMPLX_0:func.4">-</a> <a href="complex1.html#K7" title="COMPLEX1:func.7">&lt;i&gt;</a></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</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#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T22" onmouseover="rs('cfunct_1/T22')" onmouseout="rh()">CFUNCT_1:22</a></span>;<br/>
<a NAME="E7:6"/><span class="kw">then </span><span class="lab"><font color="Green" title="E10">A6</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cfunct_1.html#T26" onmouseover="rs('cfunct_1/T26')" onmouseout="rh()">CFUNCT_1:26</a></span>;<br/>
<a NAME="E8:6"/>
 <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:6"><span class="lab"><font color="Green" title="E7">A1</font></span></a>, <a class="txt" href="#E3:6"><span class="lab"><font color="Green" title="E8">A4</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" onmouseover="rs('cfdiff_1/D1')" onmouseout="rh()">CFDIFF_1:def 1</a></span>;<br/>
<a NAME="E9:6"/><span class="kw">then </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> <span class="p1">(<span class="default"><a href="comseq_2.html#K3" title="COMSEQ_2:func.3">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#K5" title="NUMBERS:func.5">0</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="ref" href="comseq_2.html#T18" onmouseover="rs('comseq_2/T18')" onmouseout="rh()">COMSEQ_2:18</a></span>;<br/>
<span class="kw">hence </span><a NAME="E10:6"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">-convergent</a>  &amp; <font color="Maroon" title="c1">seq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:6"><span class="lab"><font color="Green" title="E10">A6</font></span></a>, <a class="txt" href="#E4:6"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="ref" href="cfdiff_1.html#D1" onmouseover="rs('cfdiff_1/D1')" onmouseout="rh()">CFDIFF_1:def 1</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" 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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