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

<b>let </b><font color="Maroon">FT1</font> be  non <a href="struct_0.html#V3">empty</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">FT2</font> be  non <a href="struct_0.html#V3">empty</a> <a href="fin_topo.html#V1">filled</a>  <a href="orders_2.html#L1">RelStr</a> ;<br/><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">f</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT1</font>,<font color="Maroon">FT2</font>;<br/>







<b>assume </b><a NAME="E1:14"/>
<font color="Maroon">f</font> <a href="fintopo4.html#R2" target="_self">is_continuous</a> 0
 ;<br/>

<a NAME="E2:14"/><b>then </b><i><font color="Green">E25</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font><br/> for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  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">FT1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Olive">x</font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">y</font>,0
 
<b>by </b><i/>;<br/>
<a NAME="E3:14"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font><br/> for <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  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">FT1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> holds <br/><font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Olive">x</font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Olive">y</font>,<font color="Maroon">n</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>;<br/><b>let </b><font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>;<br/>



<b>assume </b><a NAME="E1:14_1"/><i><font color="Green">E27</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:14_1"/><i><font color="Green">E28</font></i>: 
 <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E3:14_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="fintopo3.html#K7">Finf</a> <span class="p1">(<span class="default"><a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font></span>)</span>,<font color="Maroon">n</font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#D10">FINTOPO3:def 10</a></i>;<br/>
<a NAME="E4:14_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0 <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="fintopo3.html#T36">FINTOPO3:36</a></i>;<br/>
<a NAME="E5:14_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E6:14_1"/>
<font color="Maroon">f</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0</span>)</span> <a href="tarski.html#R1">c=</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E4:14"/>
<font color="Maroon">f</font> <a href="fintopo4.html#R2" target="_self">is_continuous</a> <font color="Maroon">n</font>
 <b>by </b><i/>;<br/>


</div>
