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<div class="add">

<b>let </b><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>



<b>assume </b><a NAME="E1:25"/><i><font color="Green" title="E19">A1</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">b</font>
 ;<br/>

<b>let </b><font color="Maroon" title="c3">f</font> be  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>;<br/>

<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:25_1_1"/>
( <font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font> or <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font> )
 </span><b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_1"><b>suppose </b></a><a NAME="E1:25_1_1_1"/><i><font color="Green" title="E20">A2</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c4">L</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c5">P</font> =  <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>;<br/><b>set </b><font color="Maroon" title="c6">g</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>;<br/><a NAME="E2:25_1_1_1"/><i><font color="Green" title="E21">A3</font></i>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="treal_1.html#T18" target="_self" title="TREAL_1:th.18">Th18</a></i>;<br/><a NAME="E3:25_1_1_1"/><i><font color="Green" title="E22">A4</font></i>: 
 <a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="treal_1.html#T18" target="_self" title="TREAL_1:th.18">Th18</a></i>;<br/><i><font color="Green" title="E23">A5</font></i>: <font color="Maroon" title="c3">f</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font>
<b>by </b><i><a class="txt" href="treal_1.html#E2:25_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="tmap_1.html#T93" title="TMAP_1:th.93">TMAP_1:93</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="treal_1.html#E2:25_1_1_1"><i><font color="Green" title="E21">A3</font></i></a>, <a class="ref" href="tmap_1.html#T93" title="TMAP_1:th.93">TMAP_1:93</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
;<br/><a NAME="E5:25_1_1_1"/><i><font color="Green" title="E24">A6</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25"><i><font color="Green" title="E19">A1</font></i></a>, <a class="ref" href="treal_1.html#T11" target="_self" title="TREAL_1:th.11">Th11</a></i>;<br/><a NAME="E6:25_1_1_1"/>
 <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25_1_1_1"><i><font color="Green" title="E20">A2</font></i></a>, <a class="ref" href="treal_1.html#T15" target="_self" title="TREAL_1:th.15">Th15</a></i>;<br/><a NAME="E7:25_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/><a NAME="E8:25_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="treal_1.html#E5:25_1_1_1"><i><font color="Green" title="E24">A6</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/><b>then consider </b><font color="Maroon" title="c7">y</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span><b> such that </b><br/><a NAME="E10:25_1_1_1"/><i><font color="Green" title="E25">A7</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">y</font>
 <b>by </b><i><a class="ref" href="treal_1.html#T24" target="_self" title="TREAL_1:th.24">Th24</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_1_2"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c8">x</font> = <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p2">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>;<br/><b>thus </b><font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c8">x</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" title="TREAL_1:func.3">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p5">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p0">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p0">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p4">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="treal_1.html#E4:25_1_1_1"><i><font color="Green" title="E23">A5</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">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" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p0">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p0">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p4">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p5">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="struct_0.html#K3" title="STRUCT_0:func.3">id</a> <span class="p3">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="txt" href="treal_1.html#E3:25_1_1_1"><i><font color="Green" title="E22">A4</font></i></a>, <a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p3">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</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" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>,<span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>,<span class="p5">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c3">f</font></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span>,<span class="p4">(<span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">y</font>
<b>by </b><i><a class="ref" href="tmap_1.html#T93" title="TMAP_1:th.93">TMAP_1:93</a></i>
<br/>.= 
<font color="Maroon" title="c8">x</font>
<b>by </b><i><a class="txt" href="treal_1.html#E10:25_1_1_1"><i><font color="Green" title="E25">A7</font></i></a>, <a class="ref" href="funct_2.html#T21" title="FUNCT_2:th.21">FUNCT_2:21</a></i>
;<br/></div><b>end;</b></div><b>hence </b><a NAME="E12:25_1_1_1"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="25_1_1_2"><b>suppose </b></a><a NAME="E1:25_1_1_2"/><i><font color="Green" title="E20">A8</font></i>: 
<font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">b</font>
 ;<br/><div class="add"><a NAME="E2:25_1_1_2"/><b>then </b>
( <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c1">a</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span> &amp; <font color="Maroon" title="c1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font> )
 <b>by </b><i><a class="ref" href="treal_1.html#D1" target="_self" title="TREAL_1:def.1">Def1</a>, <a class="ref" href="rcomp_1.html#T14" title="RCOMP_1:th.14">RCOMP_1:14</a></i>;<br/><a NAME="E3:25_1_1_2"/><b>then </b><i><font color="Green" title="E21">A9</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="txt" href="treal_1.html#E1:25_1_1_2"><i><font color="Green" title="E20">A8</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="25_1_1_2_1"><b>now </b></a><div class="add"><b>take </b><font color="Maroon" title="c4">x</font> =  <a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>;<br/><b>thus </b><a NAME="E1:25_1_1_2_1"/>
<font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">x</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E3:25_1_1_2"><i><font color="Green" title="E21">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E5:25_1_1_2"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E3:25"/>
 ex <font color="Olive" title="b1">x</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font></span>)</span> st <font color="Maroon" title="c3">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">x</font>
 ;<br/>


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