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<span class="kw">let </span><font color="Maroon" title="c1">seq</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="c1">seq</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies  for <font color="Olive" title="b1">seq1</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  holds <br/> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K12" title="COMPLEX1:func.12">"</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:28"/><span class="lab"><font color="Green" title="E17">A1</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">and </span><br/><a NAME="E2:28"/><span class="lab"><font color="Green" title="E18">A2</font></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#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">seq1</font> being   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>  st <font color="Olive" title="b1">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Olive" title="b1">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  holds <br/> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">seq1</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span><br/>

<span class="kw">let </span><font color="Maroon" title="c2">seq1</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">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font> &amp; <font color="Maroon" title="c2">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a>  implies  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K12" title="COMPLEX1:func.12">"</a>  )</span><br/>

<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E3:28"/><span class="lab"><font color="Green" title="E19">A3</font></span>: 
<font color="Maroon" title="c2">seq1</font> is    <a href="valued_0.html#M2" title="VALUED_0:mode.2">subsequence</a> of <font color="Maroon" title="c1">seq</font>
 <span class="kw">and </span><br/><a NAME="E4:28"/><span class="lab"><font color="Green" title="E20">A4</font></span>: 
<font color="Maroon" title="c2">seq1</font> is  <a href="relat_1.html#V2" title="RELAT_1:attr.2">non-empty</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span><br/>

<a NAME="E5:28"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="ref" href="cfcont_1.html#T18" target="_self" title="CFCONT_1:th.18">Th40</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:28"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq1</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="complex1.html#K12" title="COMPLEX1:func.12">"</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:28"><span class="lab"><font color="Green" title="E17">A1</font></span></a>, <a class="txt" href="#E2:28"><span class="lab"><font color="Green" title="E18">A2</font></span></a>, <a class="txt" href="#E3:28"><span class="lab"><font color="Green" title="E19">A3</font></span></a>, <a class="txt" href="#E4:28"><span class="lab"><font color="Green" title="E20">A4</font></span></a>, <a class="ref" href="cfcont_1.html#T17" target="_self" title="CFCONT_1:th.17">Th39</a>, <a class="ref" href="comseq_2.html#T35" title="COMSEQ_2:th.35">COMSEQ_2:35</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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