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<span class="kw">let </span><font color="Maroon" title="c3">s9</font>, <font color="Maroon" title="c4">s</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="c3">s9</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a>  &amp; <font color="Maroon" title="c4">s</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="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c4">s</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="c3">s9</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c4">s</font></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="c3">s9</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</span> )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:50"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<font color="Maroon" title="c3">s9</font> is  <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">convergent</a> 
 <span class="kw">and </span><br/><a NAME="E2:50"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
( <font color="Maroon" title="c4">s</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="c4">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  &amp; <font color="Maroon" title="c4">s</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="Red">::  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="c3">s9</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c4">s</font></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="c3">s9</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</span></span><br/>

<a NAME="E3:50"/>
<font color="Maroon" title="c4">s</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a>  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="#E2:50"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="comseq_2.html#T34" target="_self" title="COMSEQ_2:th.34">Th34</a></span>;<br/>
<span class="kw">then </span> <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s9</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s9</font></span>)</span> <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> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s</font> <a href="valued_1.html#K36" title="VALUED_1:func.36">"</a> </span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:50"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="comseq_2.html#T30" target="_self" title="COMSEQ_2:th.30">Th30</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s9</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</span> <a href="complex1.html#K12" title="COMPLEX1:func.12">"</a> </span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:50"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="comseq_2.html#T35" target="_self" title="COMSEQ_2:th.35">Th35</a></span>
<br/>.= 
<span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s9</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E5:50"/>
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s9</font> <a href="valued_1.html#K51" title="VALUED_1:func.51">/"</a> <font color="Maroon" title="c4">s</font></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="c3">s9</font></span>)</span> <a href="complex1.html#K13" title="COMPLEX1:func.13">/</a> <span class="p1">(<span class="default"><a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c4">s</font></span>)</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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