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<span class="kw">let </span><font color="Maroon" title="c1">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_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">  for <font color="Olive" title="b1">cseq</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Olive" title="b1">cseq</font> holds <br/>( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_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="Olive" title="b1">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="Olive" title="b1">cseq</font> is  <a href="ordinal1.html#V9" title="ORDINAL1:attr.9">non-zero</a>  ) )</span><br/><span class="kw">let </span><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="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">cseq</font> implies ( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_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="E1:3"/><span class="lab"><font color="Green" title="E4">A1</font></span>: 
<font color="Maroon" title="c1">seq</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c2">cseq</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="fdiff_1.html#V1" title="FDIFF_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="E2:3"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_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="fdiff_1.html#V1" title="FDIFF_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="3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:3_1"/><span class="lab"><font color="Green" title="E5">A2</font></span>: 
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_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="E2:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A3</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> 
 
;<br/>
<a NAME="E3:3_1"/>
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E4:3_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A4</font></span>: 
 <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <font color="Maroon" title="c2">cseq</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="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a>, <a class="txt" href="cfdiff_2.html#E1"><span class="lab"><font color="Green" title="E1">Lm1</font></span></a></span>;<br/>
<a NAME="E5:3_1"/><span class="lab"><font color="Green" title="E8">A5</font></span>: 
<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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E1:3_1"><span class="lab"><font color="Green" title="E5">A2</font></span></a></span>;<br/>
<a NAME="E6:3_1"/>
<font color="Maroon" title="c2">cseq</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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E2:3_1"><span class="lab"><font color="Green" title="E6">A3</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E7:3_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="#E5:3_1"><span class="lab"><font color="Green" title="E8">A5</font></span></a>, <a class="txt" href="#E4:3_1"><span class="lab"><font color="Green" title="E7">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>; <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="E3:3"/><span class="lab"><font color="Green" title="E5">A6</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="fdiff_1.html#V1" title="FDIFF_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="E4:3"/><span class="kw">then </span>
 <a href="comseq_2.html#K3" title="COMSEQ_2:func.3">lim</a> <font color="Maroon" title="c2">cseq</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/>
<a NAME="E5:3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E6">A7</font></span>: 
 <a href="seq_2.html#K1" title="SEQ_2:func.1">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="txt" href="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="cfdiff_2.html#E2"><span class="lab"><font color="Green" title="E2">Lm2</font></span></a></span>;<br/>
<span class="kw">thus </span><a NAME="E6:3"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  <a href="fdiff_1.html#V1" title="FDIFF_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="#E1:3"><span class="lab"><font color="Green" title="E4">A1</font></span></a>, <a class="txt" href="#E3:3"><span class="lab"><font color="Green" title="E5">A6</font></span></a>, <a class="txt" href="#E5:3"><span class="lab"><font color="Green" title="E6">A7</font></span></a>, <a class="ref" href="fdiff_1.html#D1" onmouseover="rs('fdiff_1/D1')" onmouseout="rh()">FDIFF_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>
