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<b>let </b><font color="Maroon">N</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">seq</font> be   <a href="toprns_1.html#NM1">Real_Sequence</a> of <font color="Maroon">N</font>;<br/><b>let </b><font color="Maroon">seq'</font> be   <a href="toprns_1.html#NM1">Real_Sequence</a> of <font color="Maroon">N</font>;<br/>





<b>assume </b><b>that </b><br/><a NAME="E1:48"/><i><font color="Green">E44</font></i>: 
<font color="Maroon">seq</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 <b>and </b><br/><a NAME="E2:48"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">seq'</font> is <a href="toprns_1.html#V3" target="_self">convergent</a>
 ;<br/>

<b>consider </b><font color="Maroon">g1</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E4:48"/><i><font color="Green">E49</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i>, </i>;<br/>
<b>consider </b><font color="Maroon">g2</font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">N</font></span>)</span><b> such that </b><br/><a NAME="E6:48"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">r</font> being  <a href="real_1.html#NM1">Real</a>  st 0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font> holds <br/> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  st <br/>for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">n</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">r</font>
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="toprns_1.html#D9" target="_self">TOPRNS_1:def 9</a><br/>

<b>let </b><font color="Maroon">r</font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E7:48"/>
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 ;<br/>

<a NAME="E8:48"/><b>then </b><i><font color="Green">E51</font></i>: 
0 <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="seq_2.html#T3">SEQ_2:3</a></i>;<br/>
<b>then consider </b><font color="Maroon">n1</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E10:48"/><i><font color="Green">E52</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">n2</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E12:48"/><i><font color="Green">E89</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">m</font> holds <br/><span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Olive">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 <b>by </b><i>, </i>;<br/>
<b>take </b><font color="Maroon">k</font> = <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>;<br/>

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E13:48"/><i><font color="Green">E90</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 ;<br/>

<a NAME="E14:48"/>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">n1</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">n2</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E15:48"/><b>then </b>
<font color="Maroon">n1</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E16:48"/><b>then </b><i><font color="Green">E91</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i/>;<br/>
<a NAME="E17:48"/>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 
<b>by </b><i><a class="ref" href="nat_1.html#T37">NAT_1:37</a></i>;<br/>
<a NAME="E18:48"/><b>then </b>
<font color="Maroon">n2</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font>
 
<b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<a NAME="E19:48"/><b>then </b>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2
 
<b>by </b><i><a class="ref" href="toprns_1.html#T2" target="_self">Th2</a></i>;<br/>
<a NAME="E20:48"/><b>then </b><i><font color="Green">E92</font></i>: 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="real_1.html#K6">/</a> 2</span>)</span>
 
<b>by </b><i><a class="txt" href="toprns_1.html#E1:48"><i><font color="Green">E44</font></i></a>, <a class="ref" href="xreal_1.html#T10">XREAL_1:10</a></i>;<br/>
<i><font color="Green">E96</font></i>: <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="toprns_1.html#K1" target="_self">+</a> <font color="Maroon">seq'</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="euclid.html#K20">-</a> <span class="p3">(<span class="default"><font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i/>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T49">EUCLID:49</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <span class="p5">(<span class="default"><font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <span class="p4">(<span class="default"><font color="Maroon">g1</font> <a href="euclid.html#K17">+</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T50">EUCLID:50</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">g2</font></span>)</span></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p4">(<span class="default"><a href="euclid.html#K19">-</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T30">EUCLID:30</a></i>
<br/>.= 
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T45">EUCLID:45</a></i>
;<br/>
<a NAME="E22:48"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span> <a href="euclid.html#K17">+</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g1</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="real_1.html#K3">+</a> <span class="p1"><a href="toprns_1.html#K5" target="_self">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">seq'</font> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g2</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E23:48"/>
<span class="p1"><a href="toprns_1.html#K5" target="_self">|.</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">seq</font> <a href="toprns_1.html#K1" target="_self">+</a> <font color="Maroon">seq'</font></span>)</span> <a href="normsp_1.html#K2">.</a> <font color="Maroon">m</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">g</font></span>)</span></span><a href="toprns_1.html#K5" target="_self">.|</a></span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="toprns_1.html#E2:48"><i><font color="Green">E45</font></i></a>, , <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>


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