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

<b>let </b><font color="Maroon">X</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="pre_topc.html#L1">TopStruct</a> ;<br/><b>let </b><font color="Maroon">g</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a> ;<br/><b>let </b><font color="Maroon">B</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">a</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/>







<b>assume </b><a NAME="E1:4"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">p</font> where <font color="Olive">p</font> is   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> :  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Olive">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  )
 ;<br/>

<a NAME="E2:4"/>
<span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:4_1"/><i><font color="Green">E36</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<b>thus </b><a NAME="E4:4_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 <b>by </b><i>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">D</font> = <span class="p1">{<span class="default"> <font color="Olive">r</font> where <font color="Olive">r</font> is   <a href="real_1.html#NM1">Real</a> : <font color="Olive">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> </span>}</span>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E4:4"/><i><font color="Green">E38</font></i>: 
<font color="Maroon">D</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="jordan2b.html#T31">JORDAN2B:31</a></i>;<br/>
<a NAME="E5:4"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 ;<br/>

<b>then consider </b><font color="Maroon">p</font> being   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><b> such that </b><br/><a NAME="E3:4_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp;  <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 <b>by </b><i/>;<br/>
<a NAME="E4:4_2"/><i><font color="Green">E41</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E5:4_2"/><i><font color="Green">E42</font></i>: 
 <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> is   <a href="real_1.html#NM1">Real</a>
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<a NAME="E6:4_2"/>
 <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D10">JORDAN2C:def 10</a></i>;<br/>
<a NAME="E7:4_2"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i>, , </i>;<br/>
<b>hence </b><a NAME="E8:4_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:4"/>
<font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">B</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:4_3"/><i><font color="Green">E43</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 ;<br/>

<a NAME="E2:4_3"/><b>then </b>
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">D</font> )
 
<b>by </b><i><a class="ref" href="funct_1.html#D13">FUNCT_1:def 13</a></i>;<br/>
<b>then consider </b><font color="Maroon">r</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:4_3"/><i><font color="Green">E44</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">a</font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">p</font> = <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> <b>by </b><i/>;<br/>
<a NAME="E6:4_3"/>
 <a href="jordan2c.html#K6">pi</a> <font color="Maroon">g</font>,<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="jordan2c.html#D10">JORDAN2C:def 10</a></i>;<br/>
<b>hence </b><a NAME="E7:4_3"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:4"/><b>then </b>
<font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">D</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E8:4"/>
<font color="Maroon">B</font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i>, , <a class="ref" href="tops_2.html#T55">TOPS_2:55</a></i>;<br/>


</div>
