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

<span class="kw">let </span><font color="Maroon" title="c1">a</font>, <font color="Maroon" title="c2">b</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</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> holds   <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  )</span><br/>



<span class="kw">assume </span><a NAME="E1:10"/><span class="lab"><font color="Green" title="E10">A1</font></span>: 
<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>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">t1</font>, <font color="Olive" title="b2">t2</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> holds   <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Olive" title="b1">t1</font>,<font color="Olive" title="b2">t2</font>) is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> </span><br/>

<span class="kw">let </span><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> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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> </span><br/>

<span class="kw">reconsider </span><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="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">deffunc </span><font color="Maroon">H<sub>1</sub></font>(  <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>) <span class="kw">-&gt; </span>   <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>  =  <a href="subset_1.html#K10" title="SUBSET_1:func.10">In</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="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> $1</span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r1</font></span>)</span>,<a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>);<br/>
<span class="kw">consider </span><font color="Maroon" title="c7">L</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><span class="kw"> such that </span><br/><a NAME="E4:10"/><span class="lab"><font color="Green" title="E11">A2</font></span>: 
 for <font color="Olive" title="b1">r</font> being    <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>  holds  <font color="Maroon" title="c7">L</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>)
 <span class="kw">from</span> <span class="lab"><a class="ref" href="funct_2.html#S4" onmouseover="rs('funct_2/S4')" onmouseout="rh()">FUNCT_2:sch 4</a>();<br/></span>
<a NAME="E5:10"/><span class="lab"><font color="Green" title="E12">A3</font></span>: 
 for <font color="Olive" title="b1">r</font> being   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> holds  <font color="Maroon" title="c7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Olive" title="b1">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r1</font>
 
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c8">r</font> be   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r1</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c9">r</font> = <font color="Maroon" title="c8">r</font> as    <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="kw">by</span> <span class="lab"><a class="ref" href="xreal_0.html#D1" onmouseover="rs('xreal_0/D1')" onmouseout="rh()">XREAL_0:def 1</a></span>;<br/>
<a NAME="E2:10_1"/>
<font color="Maroon" title="c7">L</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c9">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c9">r</font>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10"><span class="lab"><font color="Green" title="E11">A2</font></span></a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:10_1"/>
<font color="Maroon" title="c7">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c6">r2</font> <a href="xcmplx_0.html#K6" title="XCMPLX_0:func.6">-</a> <font color="Maroon" title="c5">r1</font></span>)</span> <a href="xcmplx_0.html#K3" title="XCMPLX_0:func.3">*</a> <font color="Maroon" title="c8">r</font></span>)</span> <a href="xcmplx_0.html#K2" title="XCMPLX_0:func.2">+</a> <font color="Maroon" title="c5">r1</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">reconsider </span><font color="Maroon" title="c8">f</font> = <font color="Maroon" title="c7">L</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> <span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E7:10"/><span class="lab"><font color="Green" title="E13">A4</font></span>: 
 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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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#K3" title="TREAL_1:func.3">L[01]</a> (<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="c8">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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">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="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">w</font> holds <br/><span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">w</font></span><br/><span class="kw">let </span><font color="Maroon" title="c10">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>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font> implies <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">w</font> )</span><br/>



<span class="kw">reconsider </span><font color="Maroon" title="c11">r</font> = <font color="Maroon" title="c9">s</font> as   <a href="xreal_0.html#NM1" title="XREAL_0:NM.1">Real</a> ;<br/>
<span class="kw">assume </span><a NAME="E2:10_2"/><span class="lab"><font color="Green" title="E14">A5</font></span>: 
<font color="Maroon" title="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">w</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">w</font></span><br/>

<span class="kw">thus </span><span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> = 
<font color="Maroon">H<sub>1</sub></font>(<font color="Maroon" title="c11">r</font>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:10"><span class="lab"><font color="Green" title="E10">A1</font></span></a>, <a class="ref" href="treal_1.html#T7" target="_self" onmouseover="rs('treal_1/T7')" onmouseout="rh()">Th7</a></span>
<br/>.= 
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">w</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="txt" href="#E2:10_2"><span class="lab"><font color="Green" title="E14">A5</font></span></a></span>
; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<a NAME="E8:10"/><span class="lab"><font color="Green" title="E14">A6</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1</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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="topmetr.html#T18" onmouseover="rs('topmetr/T18')" onmouseout="rh()">TOPMETR:18</a></span>;<br/>
<a NAME="E9:10"/><span class="lab"><font color="Green" title="E15">A7</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10"><span class="lab"><font color="Green" title="E12">A3</font></span></a>, <a class="ref" href="topmetr.html#T21" onmouseover="rs('topmetr/T21')" onmouseout="rh()">TOPMETR:21</a></span>;<br/>
<a NAME="E10:10"/>
 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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> holds   <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c9">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font></span><br/>

<span class="kw">reconsider </span><font color="Maroon" title="c10">w</font> = <font color="Maroon" title="c9">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E2:10_3"/>
 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> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</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#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="tarski.html#R2" title="TARSKI: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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b2">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c11">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> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c11">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#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">G</font> implies  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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_1"/>
<font color="Maroon" title="c11">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">G</font> or  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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E3:10_3_1"/><span class="lab"><font color="Green" title="E16">A8</font></span>: 
<font color="Maroon" title="c12">G0</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E4:10_3_1"/><span class="lab"><font color="Green" title="E17">A9</font></span>: 
<font color="Maroon" title="c12">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> (<font color="Maroon" title="c1">a</font>,<font color="Maroon" title="c2">b</font>)</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">G</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tops_2.html#T24" onmouseover="rs('tops_2/T24')" onmouseout="rh()">TOPS_2:24</a></span>;<br/>
<a NAME="E5:10_3_1"/><span class="lab"><font color="Green" title="E18">A10</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="tmap_1.html#R1" title="TMAP_1:pred.1">is_continuous_at</a> <font color="Maroon" title="c10">w</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:10"><span class="lab"><font color="Green" title="E15">A7</font></span></a>, <a class="ref" href="tmap_1.html#T44" onmouseover="rs('tmap_1/T44')" onmouseout="rh()">TMAP_1:44</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:10_3_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> )</span><br/>

<a NAME="E7:10_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E13">A4</font></span></a></span>;<br/>
<a NAME="E8:10_3_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c12">G0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c13">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><span class="kw"> such that </span><br/><a NAME="E10:10_3_1"/><span class="lab"><font color="Green" title="E19">A11</font></span>: 
<font color="Maroon" title="c13">H0</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E11:10_3_1"/><span class="lab"><font color="Green" title="E20">A12</font></span>: 
<font color="Maroon" title="c10">w</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">H0</font>
 <span class="kw">and </span><br/><a NAME="E12:10_3_1"/><span class="lab"><font color="Green" title="E21">A13</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">H0</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c12">G0</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:10_3_1"><span class="lab"><font color="Green" title="E16">A8</font></span></a>, <a class="txt" href="#E5:10_3_1"><span class="lab"><font color="Green" title="E18">A10</font></span></a>, <a class="ref" href="tmap_1.html#T43" onmouseover="rs('tmap_1/T43')" onmouseout="rh()">TMAP_1:43</a></span>;<br/>
<div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_1"><span class="kw">now </span> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> )</span></a><div class="add"><span class="kw">reconsider </span><font color="Maroon" title="c14">H</font> = <font color="Maroon" title="c13">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> ;<br/><span class="kw">take </span><font color="Maroon" title="c15">H</font> = <font color="Maroon" title="c14">H</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c15">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  &amp; <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> )</span><br/><span class="kw">thus </span><a NAME="E2:10_3_1_1"/>
<font color="Maroon" title="c15">H</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:10_3_1"><span class="lab"><font color="Green" title="E19">A11</font></span></a>, <a class="ref" href="tops_2.html#T24" onmouseover="rs('tops_2/T24')" onmouseout="rh()">TOPS_2:24</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> )</span><br/><span class="kw">thus </span><a NAME="E3:10_3_1_1"/>
<font color="Maroon" title="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c15">H</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:10_3_1"><span class="lab"><font color="Green" title="E20">A12</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font></span><br/><span class="kw">thus </span><a NAME="E4:10_3_1_1"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font>
  <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><div><a class="txt" onmouseover="tooltip.show('hs2',this)" onmouseout="tooltip.hide()" onclick="hs2(this)" href="javascript:()" title="10_3_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c16">t</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a> <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font> or <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:10_3_1_1_1"/>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c15">H</font>
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font></span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c17">r</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">object</a> <span class="kw"> such that </span><br/><a NAME="E3:10_3_1_1_1"/>
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span>
 <span class="kw">and </span><br/><a NAME="E4:10_3_1_1_1"/><span class="lab"><font color="Green" title="E22">A14</font></span>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c15">H</font>
 <span class="kw">and </span><br/><a NAME="E5:10_3_1_1_1"/><span class="lab"><font color="Green" title="E23">A15</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c17">r</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<a NAME="E6:10_3_1_1_1"/><span class="lab"><font color="Green" title="E24">A16</font></span>: 
<font color="Maroon" title="c17">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_1_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c18">r</font> = <font color="Maroon" title="c17">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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_1_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a></span>;<br/>
<a NAME="E8:10_3_1_1_1"/>
<font color="Maroon" title="c18">r</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:10_3_1_1_1"><span class="lab"><font color="Green" title="E24">A16</font></span></a>, <a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E9:10_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E25">A17</font></span>: 
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:10_3_1_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c19">p</font> = <font color="Maroon" title="c18">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:10"><span class="lab"><font color="Green" title="E14">A6</font></span></a>, <a class="ref" href="tarski.html#D3" onmouseover="rs('tarski/D3')" onmouseout="rh()">TARSKI:def 3</a>, <a class="ref" href="topmetr.html#T17" onmouseover="rs('topmetr/T17')" onmouseout="rh()">TOPMETR:17</a></span>;<br/>
<a NAME="E11:10_3_1_1_1"/>
<font color="Maroon" title="c19">p</font> <a href="tarski.html#R2" title="TARSKI: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="E12:10_3_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E26">A18</font></span>: 
<font color="Maroon" title="c19">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" onmouseover="rs('funct_2/D1')" onmouseout="rh()">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E13:10_3_1_1_1"/>
( <font color="Maroon" title="c16">t</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c19">p</font> &amp; <font color="Maroon" title="c19">p</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Maroon" title="c13">H0</font> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:10"><span class="lab"><font color="Green" title="E13">A4</font></span></a>, <a class="txt" href="#E4:10_3_1_1_1"><span class="lab"><font color="Green" title="E22">A14</font></span></a>, <a class="txt" href="#E5:10_3_1_1_1"><span class="lab"><font color="Green" title="E23">A15</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>;<br/>
<a NAME="E14:10_3_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c13">H0</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:10_3_1_1_1"><span class="lab"><font color="Green" title="E26">A18</font></span></a>, <a class="ref" href="funct_1.html#D6" onmouseover="rs('funct_1/D6')" onmouseout="rh()">FUNCT_1:def 6</a></span>;<br/>
<span class="kw">hence </span><a NAME="E15:10_3_1_1_1"/>
<font color="Maroon" title="c16">t</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">G</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:10_3_1"><span class="lab"><font color="Green" title="E17">A9</font></span></a>, <a class="txt" href="#E12:10_3_1"><span class="lab"><font color="Green" title="E21">A13</font></span></a>, <a class="txt" href="#E9:10_3_1_1_1"><span class="lab"><font color="Green" title="E25">A17</font></span></a>, <a class="ref" href="xboole_0.html#D4" onmouseover="rs('xboole_0/D4')" onmouseout="rh()">XBOOLE_0:def 4</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:10_3_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> (<a href="numbers.html#K5" title="NUMBERS:func.5">0</a>,1)</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="c9">s</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b1">H</font> &amp; <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<font color="Maroon" title="c3">t1</font>,<font color="Maroon" title="c4">t2</font>)</span>)</span> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Olive" title="b1">H</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">G</font> )
 ; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E3:10_3"/>
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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="c9">s</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T43" onmouseover="rs('tmap_1/T43')" onmouseout="rh()">TMAP_1:43</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E11:10"/>
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> (<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> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tmap_1.html#T44" onmouseover="rs('tmap_1/T44')" onmouseout="rh()">TMAP_1:44</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
