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<b>let </b><font color="Maroon" title="c3">s</font>, <font color="Maroon" title="c4">s1</font> be   <a href="comseq_1.html#NM1" title="COMSEQ_1:NM.1">Complex_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c3">s</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> &amp; <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a> &amp;  <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a>  implies <font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:55"/><i><font color="Green" title="E30">A1</font></i>: 
<font color="Maroon" title="c3">s</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a>
 <b>and </b><br/><a NAME="E2:55"/><i><font color="Green" title="E31">A2</font></i>: 
<font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V1" title="COMSEQ_2:attr.1">bounded</a>
 <b>and </b><br/><a NAME="E3:55"/><i><font color="Green" title="E32">A3</font></i>: 
 <a href="comseq_2.html#K2" title="COMSEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font> is <a href="comseq_2.html#V2" title="COMSEQ_2:attr.2">convergent</a></span><br/>

<b>take </b><font color="Maroon" title="c5">g</font> =  <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="comseq_2.html#D4" target="_self" title="COMSEQ_2:def.4">COMSEQ_2:def 4</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font> holds <br/> ex <font color="Olive" title="b2">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b3">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b2">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b3">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b3">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Olive" title="b1">p</font></span><br/>

<b>let </b><font color="Maroon" title="c6">p</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> implies  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> )</span><br/>

<b>assume </b><a NAME="E4:55"/><i><font color="Green" title="E33">A4</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">n</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  st <br/>for <font color="Olive" title="b2">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Olive" title="b1">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/>

<b>consider </b><font color="Maroon" title="c7">R</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E6:55"/><i><font color="Green" title="E34">A5</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">R</font>
 <b>and </b><br/><a NAME="E7:55"/><i><font color="Green" title="E35">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   holds <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E2:55"><i><font color="Green" title="E31">A2</font></i></a>, <a class="ref" href="comseq_2.html#T8" target="_self" title="COMSEQ_2:th.8">Th8</a></i>;<br/>
<a NAME="E8:55"/><i><font color="Green" title="E36">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font>
 
<b>by </b><i><a class="txt" href="comseq_2.html#E4:55"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="comseq_2.html#E6:55"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xreal_1.html#T141" title="XREAL_1:th.141">XREAL_1:141</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c8">n1</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E10:55"/><i><font color="Green" title="E37">A8</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c8">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E1:55"><i><font color="Green" title="E30">A1</font></i></a>, <a class="txt" href="comseq_2.html#E3:55"><i><font color="Green" title="E32">A3</font></i></a>, <a class="ref" href="comseq_2.html#D5" target="_self" title="COMSEQ_2:def.5">Def5</a></i>;<br/>
<b>take </b><font color="Maroon" title="c9">n</font> = <font color="Maroon" title="c8">n1</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/><span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/>

<b>let </b><font color="Maroon" title="c10">m</font> be    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font> implies <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> )</span><br/>

<b>assume </b><a NAME="E11:55"/><i><font color="Green" title="E38">A9</font></i>: 
<font color="Maroon" title="c9">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c10">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/>

<a NAME="E12:55"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <a href="complex1.html#K5" title="COMPLEX1:func.5">0c</a> </span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 
;<br/>
<a NAME="E13:55"/><b>then </b><i><font color="Green" title="E39">A10</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font>
 
<b>by </b><i><a class="txt" href="comseq_2.html#E10:55"><i><font color="Green" title="E37">A8</font></i></a>, <a class="txt" href="comseq_2.html#E11:55"><i><font color="Green" title="E38">A9</font></i></a></i>;<br/>
<i><font color="Green" title="E40">A11</font></i>: <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> = 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K9" title="COMPLEX1:func.9">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="valued_1.html#T5" title="VALUED_1:th.5">VALUED_1:5</a></i>
<br/>.= 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
<b>by </b><i><a class="ref" href="complex1.html#T151" title="COMPLEX1:th.151">COMPLEX1:151</a></i>
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="55_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:55_2"/><i><font color="Green" title="E41">A12</font></i>: 
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <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:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font></span><br/><span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">R</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">R</font>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
<br/>.= 
<font color="Maroon" title="c6">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c7">R</font> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <font color="Maroon" title="c7">R</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c6">p</font> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> 1
<b>by </b><i><a class="txt" href="comseq_2.html#E6:55"><i><font color="Green" title="E34">A5</font></i></a>, <a class="ref" href="xcmplx_0.html#D7" title="XCMPLX_0:def.7">XCMPLX_0:def 7</a></i>
<br/>.= 
<font color="Maroon" title="c6">p</font>

;<br/><a NAME="E3:55_2"/><b>then </b><i><font color="Green" title="E42">A13</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E7:55"><i><font color="Green" title="E35">A6</font></i></a>, <a class="txt" href="comseq_2.html#E8:55"><i><font color="Green" title="E36">A7</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><a NAME="E4:55_2"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="ref" href="complex1.html#T132" title="COMPLEX1:th.132">COMPLEX1:132</a></i>;<br/><a NAME="E5:55_2"/><b>then </b>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">p</font> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <font color="Maroon" title="c7">R</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">s1</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span>
 <b>by </b><i><a class="txt" href="comseq_2.html#E13:55"><i><font color="Green" title="E39">A10</font></i></a>, <a class="txt" href="comseq_2.html#E14:55"><i><font color="Green" title="E40">A11</font></i></a>, <a class="txt" href="comseq_2.html#E1:55_2"><i><font color="Green" title="E41">A12</font></i></a>, <a class="ref" href="xreal_1.html#T70" title="XREAL_1:th.70">XREAL_1:70</a></i>;<br/><b>hence </b><a NAME="E6:55_2"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E3:55_2"><i><font color="Green" title="E42">A13</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>hence </b><a NAME="E16:55"/>
<span class="p1"><a href="complex1.html#K17" title="COMPLEX1:func.17">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">s</font> <a href="valued_1.html#K19" title="VALUED_1:func.19">(#)</a> <font color="Maroon" title="c4">s1</font></span>)</span> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c10">m</font></span>)</span> <a href="complex1.html#K11" title="COMPLEX1:func.11">-</a> <font color="Maroon" title="c5">g</font></span>)</span></span><a href="complex1.html#K17" title="COMPLEX1:func.17">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">p</font>
 <b>by </b><i><a class="txt" href="comseq_2.html#E4:55"><i><font color="Green" title="E33">A4</font></i></a>, <a class="txt" href="comseq_2.html#E14:55"><i><font color="Green" title="E40">A11</font></i></a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
