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<b>let </b><font color="Maroon" title="c1">N</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">seq</font>, <font color="Maroon" title="c3">seq'</font> be   <a href="toprns_1.html#NM1" title="TOPRNS_1:NM.1">Real_Sequence</a> of <font color="Maroon" title="c1">N</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:44"/><i><font color="Green" title="E31">A1</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 <b>and </b><br/><a NAME="E2:44"/><i><font color="Green" title="E32">A2</font></i>: 
<font color="Maroon" title="c3">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 ;<br/>

<a NAME="E3:44"/><i><font color="Green" title="E33">A3</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq'</font> is <a href="toprns_1.html#V3" title="TOPRNS_1:attr.3">convergent</a>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E1:44"><i><font color="Green" title="E31">A1</font></i></a>, <a class="txt" href="toprns_1.html#E2:44"><i><font color="Green" title="E32">A2</font></i></a>, <a class="ref" href="toprns_1.html#T41" target="_self" title="TOPRNS_1:th.41">Th41</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="44_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">r</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>assume </b><a NAME="E1:44_1"/>
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 ;<br/><a NAME="E2:44_1"/><b>then </b><i><font color="Green" title="E34">A4</font></i>: 
0 <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="ref" href="seq_2.html#T3" title="SEQ_2:th.3">SEQ_2:3</a></i>;<br/><b>then consider </b><font color="Maroon" title="c5">n1</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E4:44_1"/><i><font color="Green" title="E35">A5</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c5">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="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="toprns_1.html#E1:44"><i><font color="Green" title="E31">A1</font></i></a>, <a class="ref" href="toprns_1.html#D10" target="_self" title="TOPRNS_1:def.10">Def10</a></i>;<br/><b>consider </b><font color="Maroon" title="c6">n2</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E6:44_1"/><i><font color="Green" title="E36">A6</font></i>: 
for <font color="Olive" title="b1">m</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>   st <font color="Maroon" title="c6">n2</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="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Olive" title="b1">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:44"><i><font color="Green" title="E32">A2</font></i></a>, <a class="txt" href="toprns_1.html#E2:44_1"><i><font color="Green" title="E34">A4</font></i></a>, <a class="ref" href="toprns_1.html#D10" target="_self" title="TOPRNS_1:def.10">Def10</a></i>;<br/><b>take </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">n2</font>;<br/><b>let </b><font color="Maroon" title="c8">m</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>assume </b><a NAME="E7:44_1"/><i><font color="Green" title="E37">A7</font></i>: 
<font color="Maroon" title="c7">k</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 ;<br/><a NAME="E8:44_1"/>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c5">n1</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <font color="Maroon" title="c6">n2</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E9:44_1"/><b>then </b>
<font color="Maroon" title="c5">n1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E7:44_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E10:44_1"/><b>then </b><i><font color="Green" title="E38">A8</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="toprns_1.html#E4:44_1"><i><font color="Green" title="E35">A5</font></i></a></i>;<br/><a NAME="E11:44_1"/>
<font color="Maroon" title="c6">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T12" title="NAT_1:th.12">NAT_1:12</a></i>;<br/><a NAME="E12:44_1"/><b>then </b>
<font color="Maroon" title="c6">n2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">m</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E7:44_1"><i><font color="Green" title="E37">A7</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:44_1"/><b>then </b>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2
 <b>by </b><i><a class="txt" href="toprns_1.html#E6:44_1"><i><font color="Green" title="E36">A6</font></i></a></i>;<br/><a NAME="E14:44_1"/><b>then </b><i><font color="Green" title="E39">A9</font></i>: 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</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="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">r</font> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> 2</span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E10:44_1"><i><font color="Green" title="E38">A8</font></i></a>, <a class="ref" href="xreal_1.html#T10" title="XREAL_1:th.10">XREAL_1:10</a></i>;<br/><i><font color="Green" title="E40">A10</font></i>: <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> = 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50" title="EUCLID:th.50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="toprns_1.html#D2" target="_self" title="TOPRNS_1:def.2">Def2</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49" title="EUCLID:th.49">EUCLID:49</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p0">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p5">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p5">(<span class="default"><a href="euclid.html#K19" title="EUCLID:func.19">-</a> <span class="p0">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30" title="EUCLID:th.30">EUCLID:30</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p5">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45" title="EUCLID:th.45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49" title="EUCLID:th.49">EUCLID:49</a></i>
;<br/><a NAME="E16:44_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c3">seq'</font> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span>
 <b>by </b><i><a class="ref" href="toprns_1.html#T30" target="_self" title="TOPRNS_1:th.30">Th30</a></i>;<br/><b>hence </b><a NAME="E17:44_1"/>
<span class="p1"><a href="euclid.html#K12" title="EUCLID:func.12">|.</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="c2">seq</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="normsp_1.html#K2" title="NORMSP_1:func.2">.</a> <font color="Maroon" title="c8">m</font></span>)</span> <a href="euclid.html#K20" title="EUCLID:func.20">-</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p4">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span></span>)</span></span>)</span></span><a href="euclid.html#K12" title="EUCLID:func.12">.|</a></span> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c4">r</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E14:44_1"><i><font color="Green" title="E39">A9</font></i></a>, <a class="txt" href="toprns_1.html#E15:44_1"><i><font color="Green" title="E40">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E5:44"/>
 <a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="toprns_1.html#K1" title="TOPRNS_1:func.1">+</a> <font color="Maroon" title="c3">seq'</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="toprns_1.html#K6" title="TOPRNS_1:func.6">lim</a> <font color="Maroon" title="c3">seq'</font></span>)</span>
 <b>by </b><i><a class="txt" href="toprns_1.html#E3:44"><i><font color="Green" title="E33">A3</font></i></a>, <a class="ref" href="toprns_1.html#D10" target="_self" title="TOPRNS_1:def.10">Def10</a></i>;<br/>


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