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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>



<b>assume </b><a NAME="E1:29"/><i><font color="Green">E23</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/>

<b>let </b><font color="Maroon">c<sub>3</sub></font> be  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:29_1_1"/>
( <font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font> or <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 </span><b>by </b><i><a class="txt" href="treal_1.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_1"><b>suppose </b></a><a NAME="E1:29_1_1_1"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><b>set </b><font color="Maroon">c<sub>4</sub></font> =  <a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>5</sub></font> =  <a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span>;<br/><b>set </b><font color="Maroon">c<sub>6</sub></font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span>;<br/><a NAME="E2:29_1_1_1"/><i><font color="Green">E25</font></i>: 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="treal_1.html#T18" target="_self">Th18</a></i>;<br/><a NAME="E3:29_1_1_1"/><i><font color="Green">E26</font></i>: 
 <a href="grcat_1.html#K7">id</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="funct_2.html#R2">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="treal_1.html#T18" target="_self">Th18</a></i>;<br/><i><font color="Green">E27</font></i>: <font color="Maroon">c<sub>3</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font>
<b>by </b><i><a class="txt" href="treal_1.html#E2:29_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="treal_1.html#E2:29_1_1_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
;<br/><a NAME="E5:29_1_1_1"/><i><font color="Green">E28</font></i>: 
 <a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:29"><i><font color="Green">E23</font></i></a>, <a class="ref" href="treal_1.html#T11" target="_self">Th11</a></i>;<br/><a NAME="E6:29_1_1_1"/>
 <a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:29_1_1_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="treal_1.html#T15" target="_self">Th15</a></i>;<br/><a NAME="E7:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><a NAME="E8:29_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="txt" href="treal_1.html#E5:29_1_1_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/><b>then consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span><b> such that </b><br/><a NAME="E10:29_1_1_1"/><i><font color="Green">E29</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font>
 <b>by </b><i><a class="ref" href="treal_1.html#T24" target="_self">Th24</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>8</sub></font> = <span class="p1">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>;<br/><b>thus </b><font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>8</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p5">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p0">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="treal_1.html#E4:29_1_1_1"><i><font color="Green">E27</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p0">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p0">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p4">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="grcat_1.html#K7">id</a> <span class="p3">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="txt" href="treal_1.html#E3:29_1_1_1"><i><font color="Green">E26</font></i></a>, <a class="ref" href="relat_1.html#T55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K4" target="_self">P[01]</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">c<sub>3</sub></font></span>)</span> <a href="partfun1.html#K1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" target="_self">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="treal_1.html#K2" target="_self">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="tmap_1.html#T93">TMAP_1:93</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>
<b>by </b><i><a class="txt" href="treal_1.html#E10:29_1_1_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="funct_2.html#T21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:29_1_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="29_1_1_2"><b>suppose </b></a><a NAME="E1:29_1_1_2"/><i><font color="Green">E24</font></i>: 
<font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font>
 ;<br/><div class="add"><a NAME="E2:29_1_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default"><font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span><a href="rcomp_1.html#K1">.]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="seq_4.html#K1">{</a><span class="default"><font color="Maroon">c<sub>1</sub></font></span><a href="seq_4.html#K1">}</a></span> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="ref" href="treal_1.html#D1" target="_self">Def1</a>, <a class="ref" href="rcomp_1.html#T14">RCOMP_1:14</a></i>;<br/><a NAME="E3:29_1_1_2"/><b>then </b><i><font color="Green">E25</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span></span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:29_1_1_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="topmetr.html#T25">TOPMETR:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="29_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon">c<sub>4</sub></font> =  <a href="treal_1.html#K1" target="_self">(#)</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font>;<br/><b>thus </b><a NAME="E1:29_1_1_2_1"/>
<font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 <b>by </b><i><a class="txt" href="treal_1.html#E3:29_1_1_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:29_1_1_2"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:29"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>2</sub></font></span>)</span> st <font color="Maroon">c<sub>3</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>1</sub></font>
 ;<br/>


</div>
