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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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  iff <font color="Olive" title="b1">cseq</font> is  <a href="cfdiff_1.html#V1" title="CFDIFF_1:attr.1">convergent_to_0</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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="fdiff_1.html#V1" title="FDIFF_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="E1:4"/><span class="lab"><font color="Green" title="E5">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" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c1">seq</font> is  <a href="fdiff_1.html#V1" title="FDIFF_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="E2:4"/>
( <font color="Maroon" title="c1">seq</font> is  <a href="fdiff_1.html#V1" title="FDIFF_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="Red">::  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="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a>  )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:4_1"/><span class="lab"><font color="Green" title="E6">A2</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  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="E2:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A3</font></span>: 
<font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E3:4_1"/>
 <a href="seq_2.html#K2" title="SEQ_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="txt" href="#E1:4_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E4:4_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E8">A4</font></span>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">cseq</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="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E7">A3</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/>
<a NAME="E5:4_1"/><span class="lab"><font color="Green" title="E9">A5</font></span>: 
<font color="Maroon" title="c2">cseq</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="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E1:4_1"><span class="lab"><font color="Green" title="E6">A2</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>;<br/>
<a NAME="E6:4_1"/>
<font color="Maroon" title="c2">cseq</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="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E2:4_1"><span class="lab"><font color="Green" title="E7">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/>
<span class="kw">hence </span><a NAME="E7:4_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="#E5:4_1"><span class="lab"><font color="Green" title="E9">A5</font></span></a>, <a class="txt" href="#E4:4_1"><span class="lab"><font color="Green" title="E8">A4</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="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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

<span class="kw">assume </span><a NAME="E3:4"/><span class="lab"><font color="Green" title="E6">A6</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="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c1">seq</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> </span><br/>

<a NAME="E4:4"/><span class="kw">then </span>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">cseq</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/>
<a NAME="E5:4"/><span class="kw">then </span><span class="lab"><font color="Green" title="E7">A7</font></span>: 
 <a href="seq_2.html#K2" title="SEQ_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="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="cfdiff_2.html#E3"><span class="lab"><font color="Green" title="E3">Lm3</font></span></a></span>;<br/>
<a NAME="E6:4"/>
<font color="Maroon" title="c1">seq</font> is  <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A6</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/>
<span class="kw">hence </span><a NAME="E7:4"/>
<font color="Maroon" title="c1">seq</font> is  <a href="fdiff_1.html#V1" title="FDIFF_1:attr.1">convergent_to_0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:4"><span class="lab"><font color="Green" title="E5">A1</font></span></a>, <a class="txt" href="#E3:4"><span class="lab"><font color="Green" title="E6">A6</font></span></a>, <a class="txt" href="#E5:4"><span class="lab"><font color="Green" title="E7">A7</font></span></a>, <a class="ref" href="fdiff_1.html#D1" title="FDIFF_1:def.1">FDIFF_1:def 1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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