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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="partfun1.html#NM1">PartFunc</a> of  <a href="numbers.html#K1">REAL</a> , <a href="numbers.html#K1">REAL</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:29"/><i><font color="Green">E18</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1">compact</a>
 <b>and </b><br/><a NAME="E2:29"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R2" target="_self">is_continuous_on</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>2</sub></font> be   <a href="seq_1.html#NM1">Real_Sequence</a>;<br/><b>assume </b><a NAME="E1:29_1"/><i><font color="Green">E20</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] means ( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">a<sub>1</sub></font> );<br/><a NAME="E2:29_1"/><i><font color="Green">E21</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a>ex <font color="Olive">b<sub>2</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a>  st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font>]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>3</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/>

<a NAME="E1:29_1_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/>
<b>then consider </b><font color="Maroon">c<sub>4</sub></font> being    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K1">REAL</a> <b> such that </b><br/><a NAME="E3:29_1_1"/><i><font color="Green">E22</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:29_1"><i><font color="Green">E20</font></i></a>, <a class="ref" href="partfun1.html#T26">PARTFUN1:26</a></i>;<br/>
<b>take </b>
<font color="Maroon">c<sub>4</sub></font>
;<br/>

<b>thus </b><a NAME="E4:29_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>]
 <b>by </b><i><a class="txt" href="fcont_1.html#E3:29_1_1"><i><font color="Green">E22</font></i></a></i>;<br/>


</div><b>end;</b></div><b>consider </b><font color="Maroon">c<sub>3</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E4:29_1"/><i><font color="Green">E22</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> holds <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">b<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>]
 <b>from </b><i><a class="ref" href="rcomp_1.html#S1">RCOMP_1:sch 1</a>(<a class="txt" href="fcont_1.html#E2:29_1"><i><font color="Green">E21</font></i></a>);<br/></i><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_2"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/><b>assume </b><a NAME="E1:29_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 ;<br/><a NAME="E2:29_1_2"/><b>then </b>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="nat_1.html#NM1">Nat</a> st <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Olive">b<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="rfunct_2.html#T9">RFUNCT_2:9</a></i>;<br/><b>hence </b><a NAME="E3:29_1_2"/>
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E4:29_1"><i><font color="Green">E22</font></i></a></i>;<br/></div><b>end;</b></div><a NAME="E6:29_1"/><b>then </b><i><font color="Green">E23</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">c<sub>4</sub></font> be   <a href="nat_1.html#NM1">Nat</a>;<br/><a NAME="E1:29_1_3"/>
( <font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>3</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E4:29_1"><i><font color="Green">E22</font></i></a></i>;<br/><b>hence </b><a NAME="E2:29_1_3"/>
<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> <a href="seq_1.html#K2">.</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E6:29_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rfunct_2.html#T21">RFUNCT_2:21</a></i>;<br/></div><b>end;</b></div><a NAME="E8:29_1"/><b>then </b><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#R2">=</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="funct_2.html#T113">FUNCT_2:113</a></i>;<br/><b>consider </b><font color="Maroon">c<sub>4</sub></font> being   <a href="seq_1.html#NM1">Real_Sequence</a><b> such that </b><br/><a NAME="E10:29_1"/><i><font color="Green">E25</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E1:29"><i><font color="Green">E18</font></i></a>, <a class="txt" href="fcont_1.html#E6:29_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/><a NAME="E11:29_1"/>
<font color="Maroon">c<sub>1</sub></font> <a href="partfun1.html#K2">|</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font></span>)</span> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E2:29"><i><font color="Green">E19</font></i></a>, <a class="txt" href="fcont_1.html#E10:29_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="fcont_1.html#D2" target="_self">Def2</a></i>;<br/><a NAME="E12:29_1"/><b>then </b><i><font color="Green">E26</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="fcont_1.html#R1" target="_self">is_continuous_in</a>  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="ref" href="relat_1.html#T97">RELAT_1:97</a></i>;<br/><a NAME="E13:29_1"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E10:29_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="rfunct_2.html#T11">RFUNCT_2:11</a></i>;<br/><a NAME="E14:29_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>4</sub></font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="fcont_1.html#E6:29_1"><i><font color="Green">E23</font></i></a>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><a NAME="E15:29_1"/><b>then </b><i><font color="Green">E27</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="seq_1.html#K2">.</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="seq_2.html#K2">lim</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font></span>)</span> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E10:29_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fcont_1.html#E12:29_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="fcont_1.html#D1" target="_self">Def1</a></i>;<br/><b>take </b><font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="rfunct_2.html#K1">*</a> <font color="Maroon">c<sub>4</sub></font>;<br/><b>thus </b><a NAME="E16:29_1"/>
( <font color="Maroon">c<sub>5</sub></font> is    <a href="seqm_3.html#M1">subsequence</a> of <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>5</sub></font> is <a href="seq_2.html#V4">convergent</a> &amp;  <a href="seq_2.html#K2">lim</a> <font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> )
 <b>by </b><i><a class="txt" href="fcont_1.html#E6:29_1"><i><font color="Green">E23</font></i></a>, <a class="txt" href="fcont_1.html#E8:29_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="fcont_1.html#E10:29_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="fcont_1.html#E15:29_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a>, <a class="ref" href="rfunct_2.html#T29">RFUNCT_2:29</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E4:29"/>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> is <a href="rcomp_1.html#V1">compact</a>
 <b>by </b><i><a class="ref" href="rcomp_1.html#D3">RCOMP_1:def 3</a></i>;<br/>


</div>
