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<div class="add">

<b>let </b><font color="Maroon" title="c1">q3</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/><b>let </b><font color="Maroon" title="c2">X</font> be   <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ;<br/>



<b>assume </b><a NAME="E1:42"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c2">X</font> is <a href="rcomp_1.html#V2" title="RCOMP_1:attr.2">closed</a>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">s1</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <i><font color="Red">:: according to </font></i><a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a><br/>

<b>assume </b><b>that </b><br/><a NAME="E2:42"/><i><font color="Green" title="E28">A2</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c2">X</font>
 <b>and </b><br/><a NAME="E3:42"/><i><font color="Green" title="E29">A3</font></i>: 
<font color="Maroon" title="c3">s1</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c4">s0</font> = <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="finsop_1.html#K3" title="FINSOP_1:func.3">--&gt;</a> <font color="Maroon" title="c1">q3</font> as   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a> ;<br/>
<a NAME="E5:42"/>
<font color="Maroon" title="c4">s0</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="ref" href="seq_4.html#T39" title="SEQ_4:th.39">SEQ_4:39</a></i>;<br/>
<a NAME="E6:42"/><b>then </b><i><font color="Green" title="E30">A4</font></i>: 
<font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:42"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="seq_2.html#T25" title="SEQ_2:th.25">SEQ_2:25</a></i>;<br/>
<a NAME="E7:42"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="42_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:42_1"/><i><font color="Green" title="E31">A5</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">n</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:42_1"/><i><font color="Green" title="E32">A6</font></i>: 
<font color="Maroon" title="c6">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span>
 <b>and </b><br/><a NAME="E4:42_1"/><i><font color="Green" title="E33">A7</font></i>: 
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span> <a href="seq_1.html#K2" title="SEQ_1:func.2">.</a> <font color="Maroon" title="c6">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">n</font> = <font color="Maroon" title="c6">n</font> as    <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>by </b><i><a class="txt" href="pscomp_1.html#E3:42_1"><i><font color="Green" title="E32">A6</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">x'</font> = <font color="Maroon" title="c5">x</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="pscomp_1.html#E1:42_1"><i><font color="Green" title="E31">A5</font></i></a></i>;<br/>
<i><font color="Green" title="E34">A8</font></i>: <font color="Maroon" title="c5">x</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K9" title="SEQ_1:func.9">+</a> <span class="p2">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <font color="Maroon" title="c4">s0</font></span>)</span></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font>
<b>by </b><i><a class="txt" href="pscomp_1.html#E4:42_1"><i><font color="Green" title="E33">A7</font></i></a>, <a class="ref" href="seq_1.html#T15" title="SEQ_1:th.15">SEQ_1:15</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_1.html#K17" title="SEQ_1:func.17">-</a> <font color="Maroon" title="c4">s0</font></span>)</span> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T11" title="SEQ_1:th.11">SEQ_1:11</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c4">s0</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="seq_1.html#T14" title="SEQ_1:th.14">SEQ_1:14</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> <font color="Maroon" title="c1">q3</font></span>)</span>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">q3</font>

;<br/>
<a NAME="E8:42_1"/>
<font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c3">s1</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#T6" title="FUNCT_2:th.6">FUNCT_2:6</a></i>;<br/>
<a NAME="E9:42_1"/><b>then </b>
<font color="Maroon" title="c8">x'</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <font color="Maroon" title="c1">q3</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E2:42"><i><font color="Green" title="E28">A2</font></i></a>, <a class="txt" href="pscomp_1.html#E7:42_1"><i><font color="Green" title="E34">A8</font></i></a></i>;<br/>
<b>hence </b><a NAME="E10:42_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="ref" href="pscomp_1.html#T20" target="_self" title="PSCOMP_1:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E8:42"/><b>then </b><i><font color="Green" title="E31">A9</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="txt" href="pscomp_1.html#E1:42"><i><font color="Green" title="E27">A1</font></i></a>, <a class="txt" href="pscomp_1.html#E6:42"><i><font color="Green" title="E30">A4</font></i></a>, <a class="ref" href="rcomp_1.html#D4" title="RCOMP_1:def.4">RCOMP_1:def 4</a></i>;<br/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c4">s0</font> <a href="seq_1.html#K23" title="SEQ_1:func.23">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="txt" href="pscomp_1.html#E3:42"><i><font color="Green" title="E29">A3</font></i></a>, <a class="ref" href="seq_4.html#T59" title="SEQ_4:th.59">SEQ_4:59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s1</font></span>)</span> <a href="real_1.html#K9" title="REAL_1:func.9">-</a> <font color="Maroon" title="c1">q3</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/>
<a NAME="E10:42"/><b>then </b>
<font color="Maroon" title="c1">q3</font> <a href="real_1.html#K7" title="REAL_1:func.7">+</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c3">s1</font> <a href="seq_1.html#K10" title="SEQ_1:func.10">-</a> <font color="Maroon" title="c4">s0</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s1</font>
 
;<br/>
<b>hence </b><a NAME="E11:42"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c3">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">q3</font> <a href="measure6.html#K6" title="MEASURE6:func.6">++</a> <font color="Maroon" title="c2">X</font>
 <b>by </b><i><a class="txt" href="pscomp_1.html#E8:42"><i><font color="Green" title="E31">A9</font></i></a></i>;<br/>


</div>
