<?xml version="1.0"?>
<div class="add">

<span class="kw">let </span><font color="Maroon" title="c1">r</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</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">seq</font> being   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>  st  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; (  for <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>  holds  <font color="Olive" title="b1">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b2">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b2">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> ) holds <br/> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Olive" title="b1">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">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="firebrick">::  thesis: </font></span></a><span class="hide"> (  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font> &amp; (  for <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>  holds  <font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> ) implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:41"/><span class="lab"><font color="Green" title="E25">A1</font></span>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">and </span><br/><a NAME="E2:41"/><span class="lab"><font color="Green" title="E26">A2</font></span>: 
 for <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>  holds  <font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Olive" title="b1">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Olive" title="b1">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span><br/>

<div><span class="lab"><font color="Green" title="E27">A3</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><font color="Maroon" title="c3">p</font> be   <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>   <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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="c3">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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> )</span><br/><span class="kw">consider </span><font color="Maroon" title="c4">k1</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> <span class="kw"> such that </span><br/><a NAME="E2:41_1"/><span class="lab"><font color="Green" title="E28">A4</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a>  <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="seq_4.html#T3" target="_self" title="SEQ_4:th.3">Th10</a></span>;<br/><span class="kw">assume </span><a NAME="E3:41_1"/><span class="lab"><font color="Green" title="E29">A5</font></span>: 
 <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="c3">p</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></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/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b2">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/><a NAME="E4:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E30">A6</font></span>: 
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a></span>;<br/><a NAME="E5:41_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></span>;<br/><a NAME="E6:41_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">k1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T151" title="XREAL_1:th.151">XREAL_1:151</a></span>;<br/><a NAME="E7:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E31">A7</font></span>: 
<font color="Maroon" title="c4">k1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><span class="kw">take </span><font color="Maroon" title="c5">n</font> = <font color="Maroon" title="c4">k1</font>; <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">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="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">m</font> holds <br/> <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c6">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:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> implies  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font> )</span><br/><span class="kw">assume </span><a NAME="E8:41_1"/>
<font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font></span><br/><a NAME="E9:41_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T66" title="XREAL_1:th.66">XREAL_1:66</a></span>;<br/><a NAME="E10:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A8</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xreal_1.html#T6" title="XREAL_1:th.6">XREAL_1:6</a></span>;<br/><a NAME="E11:41_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <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="c4">k1</font> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:41_1"><span class="lab"><font color="Green" title="E28">A4</font></span></a>, <a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></span>;<br/><a NAME="E12:41_1"/><span class="kw">then </span>
<span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:41_1"><span class="lab"><font color="Green" title="E31">A7</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E13:41_1"/><span class="kw">then </span>
1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:41_1"><span class="lab"><font color="Green" title="E29">A5</font></span></a>, <a class="ref" href="xreal_1.html#T76" title="XREAL_1:th.76">XREAL_1:76</a></span>;<br/><a NAME="E14:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E33">A9</font></span>: 
1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c4">k1</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c4">k1</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> 1 <a href="binop_2.html#K11" title="BINOP_2:func.11">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c3">p</font> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a></span>)</span> <a href="binop_2.html#K8" title="BINOP_2:func.8">"</a></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/><a NAME="E15:41_1"/>
 <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="c5">n</font> <a href="square_1.html#K5" title="SQUARE_1:func.5">^2</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:41_1"><span class="lab"><font color="Green" title="E30">A6</font></span></a>, <a class="ref" href="square_1.html#T12" title="SQUARE_1:th.12">SQUARE_1:12</a></span>;<br/><a NAME="E16:41_1"/><span class="kw">then </span>
1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">n</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c5">n</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E10:41_1"><span class="lab"><font color="Green" title="E32">A8</font></span></a>, <a class="ref" href="xreal_1.html#T118" title="XREAL_1:th.118">XREAL_1:118</a></span>;<br/><a NAME="E17:41_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E34">A10</font></span>: 
1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:41_1"><span class="lab"><font color="Green" title="E33">A9</font></span></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></span>;<br/><a NAME="E18:41_1"/>
( <font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="binop_2.html#K12" title="BINOP_2:func.12">/</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K9" title="BINOP_2:func.9">+</a> <font color="Maroon" title="c1">r</font></span>)</span> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c6">m</font> <a href="nat_1.html#K4" title="NAT_1:func.4">*</a> <font color="Maroon" title="c6">m</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E26">A2</font></span></a></span>;<br/><span class="kw">hence </span><a NAME="E19:41_1"/>
 <a href="complex1.html#K18" title="COMPLEX1:func.18">abs</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="nat_1.html#K6" title="NAT_1:func.6">.</a> <font color="Maroon" title="c6">m</font></span>)</span> <a href="binop_2.html#K10" title="BINOP_2:func.10">-</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a></span>)</span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E17:41_1"><span class="lab"><font color="Green" title="E34">A10</font></span></a>, <a class="ref" href="absvalue.html#D1" title="ABSVALUE:def.1">ABSVALUE:def 1</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/></div><span class="kw">end;</span></div>
<a NAME="E4:41"/>
<font color="Maroon" title="c2">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="#E1:41"><span class="lab"><font color="Green" title="E25">A1</font></span></a>, <a class="txt" href="#E2:41"><span class="lab"><font color="Green" title="E26">A2</font></span></a>, <a class="ref" href="seq_4.html#T32" target="_self" title="SEQ_4:th.32">Th47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:41"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">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="#E3:41"><span class="lab"><font color="Green" title="E27">A3</font></span></a>, <a class="ref" href="seq_2.html#D7" title="SEQ_2:def.7">SEQ_2:def 7</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/>


</div>
