<?xml version="1.0"?>
<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="E1:15"/><i><font color="Green" title="E13">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">t1</font>, <font color="Maroon" title="c4">t2</font> be   <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>;<br/>

<b>reconsider </b><font color="Maroon" title="c5">r1</font> = <font color="Maroon" title="c3">t1</font>, <font color="Maroon" title="c6">r2</font> = <font color="Maroon" title="c4">t2</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="treal_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c7">a</font> = <font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c8">b</font> = <font color="Maroon" title="c2">b</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="ref" href="xreal_0.html#D1" title="XREAL_0:def.1">XREAL_0:def 1</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> ) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> $1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c9">P</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of  <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a> <b> such that </b><br/><a NAME="E5:15"/><i><font color="Green" title="E14">A2</font></i>: 
for <font color="Olive" title="b1">r</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> holds  <font color="Maroon" title="c9">P</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">r</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4" title="FUNCT_2:sch.4">FUNCT_2:sch 4</a>();<br/></i>
<b>reconsider </b><font color="Maroon" title="c10">f</font> = <font color="Maroon" title="c9">P</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E7:15"/><i><font color="Green" title="E15">A3</font></i>: 
<font color="Maroon" title="c10">f</font> 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#E5:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="ref" href="topmetr.html#T28" title="TOPMETR:th.28">TOPMETR:28</a></i>;<br/>
<a NAME="E8:15"/><i><font color="Green" title="E16">A4</font></i>: 
for <font color="Olive" title="b1">s</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span><br/> for <font color="Olive" title="b2">w</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">w</font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b2">w</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be   <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c12">w</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>



<b>reconsider </b><font color="Maroon" title="c13">r</font> = <font color="Maroon" title="c11">s</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="treal_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="txt" href="treal_1.html#E2"><i><font color="Green" title="E2">Lm2</font></i></a></i>;<br/>
<b>assume </b><a NAME="E2:15_1"/><i><font color="Green" title="E17">A5</font></i>: 
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">w</font>
 ;<br/>

<b>thus </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="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p3">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c13">r</font></span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c7">a</font> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> <font color="Maroon" title="c6">r2</font></span>)</span></span>)</span> <a href="real_1.html#K6" title="REAL_1:func.6">/</a> <span class="p2">(<span class="default"><font color="Maroon" title="c8">b</font> <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c7">a</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="treal_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="treal_1.html#T14" target="_self" title="TREAL_1:th.14">Th14</a></i>
<br/>.= 
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">w</font>
<b>by </b><i><a class="txt" href="treal_1.html#E5:15"><i><font color="Green" title="E14">A2</font></i></a>, <a class="txt" href="treal_1.html#E2:15_1"><i><font color="Green" title="E17">A5</font></i></a></i>
;<br/>


</div><b>end;</b></div>
<a NAME="E9:15"/><i><font color="Green" title="E17">A6</font></i>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">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> 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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E1:15"><i><font color="Green" title="E13">A1</font></i></a>, <a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a></i>;<br/>
<a NAME="E10:15"/>
for <font color="Olive" title="b1">s</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> holds   <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Olive" title="b1">s</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">s</font> be   <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">w</font> = <font color="Maroon" title="c11">s</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="treal_1.html#E9:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E2:15_2"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">G</font> holds <br/> ex <font color="Olive" title="b2">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> st <br/>( <font color="Olive" title="b2">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b2">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">G</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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>;<br/>

<b>assume </b><a NAME="E1:15_2_1"/>
<font color="Maroon" title="c13">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c14">G0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E3:15_2_1"/><i><font color="Green" title="E18">A7</font></i>: 
<font color="Maroon" title="c14">G0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E4:15_2_1"/><i><font color="Green" title="E19">A8</font></i>: 
<font color="Maroon" title="c14">G0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">G</font>
 <b>by </b><i><a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>assume </b><a NAME="E5:15_2_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">G</font>
 ;<br/>

<a NAME="E6:15_2_1"/><b>then </b>
<font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">G</font>
 
<b>by </b><i><a class="txt" href="treal_1.html#E8:15"><i><font color="Green" title="E16">A4</font></i></a></i>;<br/>
<a NAME="E7:15_2_1"/><b>then </b>
( <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c14">G0</font> &amp; <font color="Maroon" title="c10">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c12">w</font> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E7:15"><i><font color="Green" title="E15">A3</font></i></a>, <a class="txt" href="treal_1.html#E4:15_2_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">H0</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> <b> such that </b><br/><a NAME="E9:15_2_1"/><i><font color="Green" title="E20">A9</font></i>: 
<font color="Maroon" title="c15">H0</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>and </b><br/><a NAME="E10:15_2_1"/><i><font color="Green" title="E21">A10</font></i>: 
<font color="Maroon" title="c12">w</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">H0</font>
 <b>and </b><br/><a NAME="E11:15_2_1"/><i><font color="Green" title="E22">A11</font></i>: 
<font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c14">G0</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E3:15_2_1"><i><font color="Green" title="E18">A7</font></i></a>, <a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1"><b>now </b></a><div class="add"><b>reconsider </b><font color="Maroon" title="c16">H</font> = <font color="Maroon" title="c15">H0</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> ;<br/><b>take </b><font color="Maroon" title="c17">H</font> = <font color="Maroon" title="c16">H</font>;<br/><b>thus </b><a NAME="E2:15_2_1_1"/>
<font color="Maroon" title="c17">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="treal_1.html#E9:15_2_1"><i><font color="Green" title="E20">A9</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/><b>thus </b><a NAME="E3:15_2_1_1"/>
<font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E10:15_2_1"><i><font color="Green" title="E21">A10</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/><b>thus </b><a NAME="E4:15_2_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c17">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">G</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">t</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="E1:15_2_1_1_1"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c17">H</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:15_2_1_1_1"/>
<font color="Maroon" title="c19">r</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"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span>
 <b>and </b><br/><a NAME="E4:15_2_1_1_1"/><i><font color="Green" title="E23">A12</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c17">H</font>
 <b>and </b><br/><a NAME="E5:15_2_1_1_1"/><i><font color="Green" title="E24">A13</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.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="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">r</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E6:15_2_1_1_1"/><i><font color="Green" title="E25">A14</font></i>: 
<font color="Maroon" title="c19">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E4:15_2_1_1_1"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">r</font> = <font color="Maroon" title="c19">r</font> as   <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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> <b>by </b><i><a class="txt" href="treal_1.html#E4:15_2_1_1_1"><i><font color="Green" title="E23">A12</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">p</font> = <font color="Maroon" title="c20">r</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="txt" href="treal_1.html#E9:15"><i><font color="Green" title="E17">A6</font></i></a>, <a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E9:15_2_1_1_1"/>
<font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> 
 
;<br/>
<a NAME="E10:15_2_1_1_1"/><b>then </b>
( <font color="Maroon" title="c18">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c21">p</font> &amp; <font color="Maroon" title="c21">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">H0</font> &amp; <font color="Maroon" title="c21">p</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> <font color="Maroon" title="c10">f</font> )
 
<b>by </b><i><a class="txt" href="treal_1.html#E8:15"><i><font color="Green" title="E16">A4</font></i></a>, <a class="txt" href="treal_1.html#E4:15_2_1_1_1"><i><font color="Green" title="E23">A12</font></i></a>, <a class="txt" href="treal_1.html#E5:15_2_1_1_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E11:15_2_1_1_1"/><b>then </b><i><font color="Green" title="E26">A15</font></i>: 
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c15">H0</font>
 
<b>by </b><i><a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E12:15_2_1_1_1"/>
<font color="Maroon" title="c20">r</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"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E6:15_2_1_1_1"><i><font color="Green" title="E25">A14</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E13:15_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</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="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> 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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span>
 
<b>by </b><i><a class="txt" href="treal_1.html#E5:15_2_1_1_1"><i><font color="Green" title="E24">A13</font></i></a>, <a class="ref" href="funct_1.html#D12" title="FUNCT_1:def.12">FUNCT_1:def 12</a></i>;<br/>
<b>hence </b><a NAME="E14:15_2_1_1_1"/>
<font color="Maroon" title="c18">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">G</font>
 <b>by </b><i><a class="txt" href="treal_1.html#E4:15_2_1"><i><font color="Green" title="E19">A8</font></i></a>, <a class="txt" href="treal_1.html#E11:15_2_1"><i><font color="Green" title="E22">A11</font></i></a>, <a class="txt" href="treal_1.html#E11:15_2_1_1_1"><i><font color="Green" title="E26">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E13:15_2_1"/>
 ex <font color="Olive" title="b1">H</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="c7">a</font>,<font color="Maroon" title="c8">b</font></span>)</span> st <br/>( <font color="Olive" title="b1">H</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c11">s</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font></span>)</span> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c13">G</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:15_2"/>
 <a href="treal_1.html#K4" title="TREAL_1:func.4">P[01]</a> <font color="Maroon" title="c7">a</font>,<font color="Maroon" title="c8">b</font>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c11">s</font>
 <b>by </b><i><a class="ref" href="tmap_1.html#T48" title="TMAP_1:th.48">TMAP_1:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E11:15"/>
 <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>,<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font> 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="ref" href="tmap_1.html#T49" title="TMAP_1:th.49">TMAP_1:49</a></i>;<br/>


</div>
