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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">P</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">w1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w3</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w4</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w5</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w6</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">w7</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">w1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w3</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w4</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w5</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w6</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp; <font color="Maroon">w7</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w1</font>,<font color="Maroon">w2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w2</font>,<font color="Maroon">w3</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w3</font>,<font color="Maroon">w4</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w4</font>,<font color="Maroon">w5</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w5</font>,<font color="Maroon">w6</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> &amp;  <a href="topreal1.html#K3">LSeg</a> <font color="Maroon">w6</font>,<font color="Maroon">w7</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P</font> )
 ;<br/>

<a NAME="E2:51"/><b>then </b><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">h2</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> st <br/>( <font color="Olive">h2</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">h2</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Olive">h2</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i/>;<br/>
<a NAME="E3:51"/>
 ex <font color="Olive">h4</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> st <br/>( <font color="Olive">h4</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">w4</font> <a href="hidden.html#R1">=</a> <font color="Olive">h4</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">h4</font> <a href="funct_1.html#K1">.</a> 1 )
 
<b>by </b><i>, </i>;<br/>
<b>hence </b><a NAME="E4:51"/>
 ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P</font></span>)</span> st <br/>( <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">w1</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">w7</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, , </i>;<br/>


</div>
