<?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> , <font color="Maroon">FT2</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">h</font> be   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT1</font>,<font color="Maroon">FT2</font>;<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">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="E1:5"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">h</font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> )
 ;<br/>

<a NAME="E2:5"/>
<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>
 
;<br/>
<a NAME="E3:5"/><b>then </b><i><font color="Green">E33</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">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:5"/><i><font color="Green">E35</font></i>: 
( <font color="Maroon">h</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">h</font> is <a href="funct_2.html#V2">onto</a> &amp; ( for <font color="Olive">x2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> holds  <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x2</font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i>, </i>;<br/>
<b>let </b><font color="Maroon">z</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>;<br/>

<a NAME="E5:5"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT1</font>
 
;<br/>
<a NAME="E6:5"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><i><font color="Green">E41</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_1"/><i><font color="Green">E42</font></i>: 
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><a NAME="E2:5_1"/>
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0 holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:5_1_1"/><i><font color="Green">E45</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i/>;<br/>
<a NAME="E2:5_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT1</font>
 
;<br/>
<a NAME="E3:5_1_1"/><b>then </b><i><font color="Green">E46</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0
 ;<br/><a NAME="E2:5_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E3:5_1_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><a NAME="E4:5_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">x</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>;<br/><a NAME="E5:5_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font>
 <b>by </b><i>, <a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, </i>;<br/><b>hence </b><a NAME="E6:5_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</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/></div>
<b>end;</b></div>


</div><b>end;</b></div><a NAME="E3:5_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E4:5_1"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_1_2"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:5_1_2"/>
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_1_2_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:5_1_2_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i><a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>;<br/>
<a NAME="E3:5_1_2_1"/><b>then </b>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x3</font> where <font color="Olive">x3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> :  ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font> &amp; <font color="Olive">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#D14">FIN_TOPO:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">x3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font><b> such that </b><br/><a NAME="E5:5_1_2_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y3</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font><b> such that </b><br/><a NAME="E7:5_1_2_1"/><i><font color="Green">E56</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font>, <font color="Maroon">p</font> = <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x3</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> ;<br/>
<a NAME="E9:5_1_2_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">w2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>  st <font color="Olive">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0 holds <br/><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Olive">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_1_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font>;<br/>

<a NAME="E1:5_1_2_1_1"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:5_1_2_1_1"/>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT1</font>
 
;<br/>
<a NAME="E3:5_1_2_1_1"/><b>then </b><i><font color="Green">E62</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_1_2_1_1_1"/>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0
 ;<br/><a NAME="E2:5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E3:5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><a NAME="E4:5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <font color="Maroon">h</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, <a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/><a NAME="E5:5_1_2_1_1_1"/><b>then </b>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, </i>;<br/><b>hence </b><a NAME="E6:5_1_2_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E10:5_1_2_1"/><i><font color="Green">E63</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E11:5_1_2_1"/>
<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0
 
<b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E12:5_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 
<b>by </b><i/>;<br/>
<a NAME="E13:5_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E14:5_1_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T16">FIN_TOPO:16</a>, </i>;<br/>
<b>hence </b><a NAME="E15:5_1_2_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E5:5_1"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><b>hence </b><a NAME="E6:5_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/></div><b>end;</b></div>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:5_2"/><i><font color="Green">E64</font></i>: 
<font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font>
 ;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">a<sub>1</sub></font> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">a<sub>1</sub></font>;<br/><b>consider </b><font color="Maroon">g</font> being   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">FT2</font>,<font color="Maroon">FT1</font><b> such that </b><br/><a NAME="E3:5_2"/><i><font color="Green">E65</font></i>: 
( <font color="Maroon">g</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K2">"</a>  &amp; <font color="Maroon">g</font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a>  )
 <b>by </b><i>, </i>;<br/><a NAME="E4:5_2"/><i><font color="Green">E66</font></i>: 
( <font color="Maroon">g</font> <a href="fintopo5.html#R1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> )
 <b>by </b><i>, , <a class="ref" href="funct_1.html#T56">FUNCT_1:56</a>, , </i>;<br/><a NAME="E5:5_2"/>
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0 holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_1"><b>proof </b></a><div class="add">

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

<a NAME="E1:5_2_1"/><i><font color="Green">E67</font></i>: 
<font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E2:5_2_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT2</font>
 
;<br/>
<a NAME="E3:5_2_1"/><b>then </b><i><font color="Green">E68</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_2_1_1"/>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,0
 ;<br/><a NAME="E2:5_2_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E3:5_2_1_1"/><b>then </b>
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><a NAME="E4:5_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a> <font color="Maroon">g</font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, </i>;<br/><a NAME="E5:5_2_1_1"/><b>then </b>
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">x</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, , </i>;<br/><b>hence </b><a NAME="E6:5_2_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,0
 <b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div><a NAME="E6:5_2"/><b>then </b><i><font color="Green">E69</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 ;<br/><a NAME="E7:5_2"/><i><font color="Green">E70</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:5_2_2"/><i><font color="Green">E71</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<a NAME="E2:5_2_2"/>
for <font color="Olive">w</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  st <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:5_2_2_1"/><i><font color="Green">E72</font></i>: 
<font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 ;<br/>

<a NAME="E2:5_2_2_1"/>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i><a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>;<br/>
<a NAME="E3:5_2_2_1"/><b>then </b>
 <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">x3</font> where <font color="Olive">x3</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> :  ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font> &amp; <font color="Olive">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y3</font> ) </span>}</span> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#D14">FIN_TOPO:def 14</a></i>;<br/>
<b>then consider </b><font color="Maroon">x3</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font><b> such that </b><br/><a NAME="E5:5_2_2_1"/><i><font color="Green">E73</font></i>: 
( <font color="Maroon">x3</font> <a href="hidden.html#R1">=</a> <font color="Maroon">w</font> &amp;  ex <font color="Olive">y3</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> st <br/>( <font color="Olive">y3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Olive">y3</font> ) )
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font><b> such that </b><br/><a NAME="E7:5_2_2_1"/><i><font color="Green">E74</font></i>: 
( <font color="Maroon">y2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">k</font> &amp; <font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font> )
 <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">q</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">y2</font>, <font color="Maroon">p</font> = <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x3</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT1</font> ;<br/>
<a NAME="E9:5_2_2_1"/><i><font color="Green">E75</font></i>: 
( <font color="Maroon">g</font> is <a href="funct_1.html#V2">one-to-one</a> &amp; <font color="Maroon">g</font> is <a href="funct_2.html#V2">onto</a> &amp; ( for <font color="Olive">x4</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font> holds  <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Olive">x4</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">x4</font></span>)</span></span><a href="tarski.html#K1">}</a></span> ) )
 
<b>by </b><i>, </i>;<br/>
<a NAME="E10:5_2_2_1"/><i><font color="Green">E77</font></i>: 
for <font color="Olive">w2</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>  st <font color="Olive">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0 holds <br/><span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Olive">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="5_2_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">w2</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">FT2</font>;<br/>

<a NAME="E1:5_2_2_1_1"/><i><font color="Green">E78</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT1</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">y2</font></span>)</span></span><a href="tarski.html#K1">}</a></span>
 
<b>by </b><i><a class="ref" href="fintopo5.html#T1" target="_self">Th1</a></i>;<br/>
<a NAME="E2:5_2_2_1_1"/>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">FT2</font>
 
;<br/>
<a NAME="E3:5_2_2_1_1"/><b>then </b><i><font color="Green">E79</font></i>: 
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:5_2_2_1_1_1"/>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0
 ;<br/><a NAME="E2:5_2_2_1_1_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">y2</font>
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/><a NAME="E3:5_2_2_1_1_1"/><b>then </b>
<font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a></i>;<br/><a NAME="E4:5_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="relat_1.html#K9">.:</a> <span class="p1">(<span class="default">the <a href="orders_2.html#U1">InternalRel</a> of <font color="Maroon">FT2</font> <a href="relset_1.html#K10">.:</a> <span class="p2"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">y2</font></span><a href="tarski.html#K1">}</a></span></span>)</span>
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a>, , </i>;<br/><a NAME="E5:5_2_2_1_1_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">q</font>
 <b>by </b><i><a class="ref" href="fin_topo.html#D1">FIN_TOPO:def 1</a>, , </i>;<br/><b>hence </b><a NAME="E6:5_2_2_1_1_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w2</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 <b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/></div>
<b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E11:5_2_2_1"/><i><font color="Green">E80</font></i>: 
<font color="Maroon">q</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font>
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E12:5_2_2_1"/>
<font color="Maroon">x3</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y2</font>,0
 
<b>by </b><i>, <a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E13:5_2_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">q</font>,0
 
<b>by </b><i>, </i>;<br/>
<a NAME="E14:5_2_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="fin_topo.html#K1">U_FT</a> <font color="Maroon">q</font>
 
<b>by </b><i><a class="ref" href="fintopo3.html#T47">FINTOPO3:47</a></i>;<br/>
<a NAME="E15:5_2_2_1"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">k</font></span>)</span> <a href="fin_topo.html#K12">^f</a> 
 
<b>by </b><i><a class="ref" href="fin_topo.html#T16">FIN_TOPO:16</a>, </i>;<br/>
<b>hence </b><a NAME="E16:5_2_2_1"/>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">w</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, , <a class="ref" href="fintopo3.html#T48">FINTOPO3:48</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:5_2_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/>


</div><b>end;</b></div><a NAME="E8:5_2"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">k</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E9:5_2"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:5_2"/>
<font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">n</font>
 <b>by </b><i><a class="ref" href="funct_1.html#T56">FUNCT_1:56</a>, , </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E9:5"/>
( <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">x</font>,<font color="Maroon">n</font> iff <font color="Maroon">h</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">z</font> <a href="hidden.html#R2">in</a>  <a href="fintopo3.html#K10">U_FT</a> <font color="Maroon">y</font>,<font color="Maroon">n</font> )
 <b>by </b><i/>;<br/>


</div>
