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

<b>let </b><font color="Maroon" title="c1">X</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>let </b><font color="Maroon" title="c2">f</font> be   <a href="partfun1.html#NM1" title="PARTFUN1:NM.1">PartFunc</a> of  <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> , <a href="supinf_1.html#K3" title="SUPINF_1:func.3">REAL</a> ;<br/>



<a NAME="E1:31"/><i><font color="Green" title="E23">A1</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R6" title="RFUNCT_4:pred.6">is_upper_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R8" title="RFUNCT_4:pred.8">is_lower_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> implies <font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_1"/><i><font color="Green" title="E24">A2</font></i>: 
( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R6" title="RFUNCT_4:pred.6">is_upper_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R8" title="RFUNCT_4:pred.8">is_lower_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> )
 ;<br/>

<a NAME="E2:31_1"/>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; ( for <font color="Olive" title="b1">x0</font> being <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a>   st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Olive" title="b1">x0</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_1_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_1_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:31_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="ref" href="rfunct_4.html#D6" target="_self" title="RFUNCT_4:def.6">Def6</a></i>;<br/>

<b>let </b><font color="Maroon" title="c3">x0</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="E2:31_1_1"/><i><font color="Green" title="E25">A3</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E3:31_1_1"/>
<font color="Maroon" title="c3">x0</font> is   <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/>
<a NAME="E4:31_1_1"/><b>then </b>
( <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font> &amp; <font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font> )
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:31_1"><i><font color="Green" title="E24">A2</font></i></a>, <a class="txt" href="rfunct_4.html#E2:31_1_1"><i><font color="Green" title="E25">A3</font></i></a>, <a class="ref" href="rfunct_4.html#D6" target="_self" title="RFUNCT_4:def.6">Def6</a>, <a class="ref" href="rfunct_4.html#D8" target="_self" title="RFUNCT_4:def.8">Def8</a></i>;<br/>
<b>hence </b><a NAME="E5:31_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="ref" href="rfunct_4.html#T20" target="_self" title="RFUNCT_4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_1"/>
<font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 <b>by </b><i><a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:31"/>
( <font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font> implies ( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R6" title="RFUNCT_4:pred.6">is_upper_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R8" title="RFUNCT_4:pred.8">is_lower_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:31_2"/><i><font color="Green" title="E24">A4</font></i>: 
<font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:31_2"/>
( <font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font> &amp; ( for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Olive" title="b1">x0</font> ) &amp; ( for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Olive" title="b1">x0</font> ) )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:31_2_1"/>
<font color="Maroon" title="c1">X</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <font color="Maroon" title="c2">f</font>
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:31_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>

<a NAME="E2:31_2_1"/><i><font color="Green" title="E25">A5</font></i>: 
for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:31_2_1_1"/><i><font color="Green" title="E26">A6</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:31_2_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:31_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="rfunct_4.html#E1:31_2_1_1"><i><font color="Green" title="E26">A6</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:31_2_1_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="ref" href="rfunct_4.html#T20" target="_self" title="RFUNCT_4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_2_1_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E3:31_2_1"/>
for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Olive" title="b1">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">x0</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>

<b>assume </b><a NAME="E1:31_2_1_2"/><i><font color="Green" title="E26">A7</font></i>: 
<font color="Maroon" title="c3">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font>
 ;<br/>

<a NAME="E2:31_2_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="31_2_1_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:31_2_1_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="fcont_1.html#R1" title="FCONT_1:pred.1">is_continuous_in</a> <font color="Maroon" title="c3">x0</font>
 
<b>by </b><i><a class="txt" href="rfunct_4.html#E1:31_2"><i><font color="Green" title="E24">A4</font></i></a>, <a class="txt" href="rfunct_4.html#E1:31_2_1_2"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="fcont_1.html#D2" title="FCONT_1:def.2">FCONT_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E2:31_2_1_2_1"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 <b>by </b><i><a class="ref" href="rfunct_4.html#T20" target="_self" title="RFUNCT_4:th.20">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_2_1_2"/>
<font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Maroon" title="c3">x0</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:31_2_1"/>
( ( for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R5" title="RFUNCT_4:pred.5">is_upper_semicontinuous_in</a> <font color="Olive" title="b1">x0</font> ) &amp; ( for <font color="Olive" title="b1">x0</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>  st <font color="Olive" title="b1">x0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">X</font> holds <br/><font color="Maroon" title="c2">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c1">X</font> <a href="rfunct_4.html#R7" title="RFUNCT_4:pred.7">is_lower_semicontinuous_in</a> <font color="Olive" title="b1">x0</font> ) )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E2:31_2_1"><i><font color="Green" title="E25">A5</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31_2"/>
( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R6" title="RFUNCT_4:pred.6">is_upper_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R8" title="RFUNCT_4:pred.8">is_lower_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="ref" href="rfunct_4.html#D6" target="_self" title="RFUNCT_4:def.6">Def6</a>, <a class="ref" href="rfunct_4.html#D8" target="_self" title="RFUNCT_4:def.8">Def8</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:31"/>
( ( <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R6" title="RFUNCT_4:pred.6">is_upper_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> &amp; <font color="Maroon" title="c2">f</font> <a href="rfunct_4.html#R8" title="RFUNCT_4:pred.8">is_lower_semicontinuous_on</a> <font color="Maroon" title="c1">X</font> ) iff <font color="Maroon" title="c2">f</font> <a href="fcont_1.html#R2" title="FCONT_1:pred.2">is_continuous_on</a> <font color="Maroon" title="c1">X</font> )
 <b>by </b><i><a class="txt" href="rfunct_4.html#E1:31"><i><font color="Green" title="E23">A1</font></i></a></i>;<br/>


</div>
