<?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#NM1">TopSpace</a>;<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>set </b><font color="Maroon">g1</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font>;<br/>
<a NAME="E1:26"/><i><font color="Green">E27</font></i>: 
(  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> &amp; ( for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> holds <br/><span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) )
 
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a>, <a class="ref" href="funcop_1.html#T19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E2:26"/>
 <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&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="26_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:26_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_1"/><i><font color="Green">E28</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E4:26_1"/>
<font color="Maroon">a</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 
<b>by </b><i><a class="ref" href="xreal_0.html#D1">XREAL_0:def 1</a></i>;<br/>
<b>hence </b><a NAME="E5:26_1"/>
<font color="Maroon">y</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="txt" href="jgraph_2.html#E1:26"><i><font color="Green">E27</font></i></a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">g0</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i>, <a class="ref" href="funct_2.html#T4">FUNCT_2:4</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <span class="p1">(<span class="default">the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="funcop_1.html#K2">--&gt;</a> <font color="Maroon">a</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i/>;<br/>
<a NAME="E5:26"/>
for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font><br/> for <font color="Olive">V</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">V</font> &amp; <font color="Olive">V</font> is <a href="pre_topc.html#V3">open</a> holds <br/> ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Olive">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Olive">V</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">p</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font>;<br/><b>let </b><font color="Maroon">V</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>



<b>assume </b><a NAME="E1:26_2"/><i><font color="Green">E43</font></i>: 
( <font color="Maroon">g0</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font> &amp; <font color="Maroon">V</font> is <a href="pre_topc.html#V3">open</a> )
 ;<br/>

<b>set </b><font color="Maroon">f1</font> = <font color="Maroon">g0</font>;<br/>
<b>set </b><font color="Maroon">G1</font> = <font color="Maroon">V</font>;<br/>
<a NAME="E2:26_2"/><i><font color="Green">E44</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> is <a href="pre_topc.html#V3">open</a>
 
<b>by </b><i><a class="ref" href="tops_1.html#T53">TOPS_1:53</a></i>;<br/>
<a NAME="E3:26_2"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">y</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="E1:26_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:26_2_1"/><i><font color="Green">E46</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g0</font> &amp; <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g0</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<a NAME="E4:26_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font>
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E5:26_2_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">V</font>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div>
<a NAME="E4:26_2"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font>
 
;<br/>
<a NAME="E5:26_2"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <font color="Maroon">X</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>hence </b><a NAME="E6:26_2"/>
 ex <font color="Olive">W</font> being  <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">X</font> st <br/>( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Olive">W</font> &amp; <font color="Olive">W</font> is <a href="pre_topc.html#V3">open</a> &amp; <font color="Maroon">g0</font> <a href="relset_1.html#K10">.:</a> <font color="Olive">W</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">V</font> )
 <b>by </b><i><a class="ref" href="jgraph_2.html#T6" target="_self">Th6</a>, <a class="ref" href="jgraph_2.html#T8" target="_self">Th8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E6:26"/><b>then </b>
<font color="Maroon">g0</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E7:26"/>
 ex <font color="Olive">g</font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">X</font>,<a href="topmetr.html#K3">R^1</a>  st <br/>( ( for <font color="Olive">p</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">X</font> holds  <font color="Olive">g</font> <a href="funct_2.html#K3">.</a> <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> ) &amp; <font color="Olive">g</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="jgraph_2.html#E1:26"><i><font color="Green">E27</font></i></a></i>;<br/>


</div>
