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

<b>let </b><font color="Maroon">n</font> be  non <a href="xboole_0.html#V1">empty</a>  <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xxreal_0.html#V2">positive</a> <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">x</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>set </b><font color="Maroon">U</font> =  <a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font>;<br/>
<b>set </b><font color="Maroon">C</font> =  <a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span>,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">w</font> being  <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> st <br/>( <font color="Olive">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Olive">w</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font> );<br/>
<a NAME="E1:53"/><i><font color="Green">E61</font></i>: 
<font color="Maroon">r</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="E2:53"/><i><font color="Green">E66</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="topreal9.html#K3">Sphere</a> <font color="Maroon">x</font>,<font color="Maroon">r</font>
 
<b>by </b><i/>;<br/>
<a NAME="E3:53"/><i><font color="Green">E68</font></i>: 
for <font color="Olive">u</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span>  ex <font color="Olive">y</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">u</font>,<font color="Olive">y</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">u</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span>;<br/>

<b>reconsider </b><font color="Maroon">v</font> = <font color="Maroon">u</font> as   <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> <b>by </b><i><a class="ref" href="topmetr.html#T2">TOPMETR:2</a></i>;<br/>
<b>set </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">v</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font>;<br/>
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">v</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font></span>)</span> <a href="euclid.html#K20">-</a> <font color="Maroon">x</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span> = 
<span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">v</font></span>)</span></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="euclid.html#T52">EUCLID:52</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="complex1.html#K18">abs</a> <font color="Maroon">r</font></span>)</span> <a href="real_1.html#K4">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">v</font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i>, <a class="ref" href="toprns_1.html#T8">TOPRNS_1:8</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> <span class="p1"><a href="toprns_1.html#K5">|.</a><span class="default"><font color="Maroon">v</font></span><a href="toprns_1.html#K5">.|</a></span>
<b>by </b><i><a class="ref" href="absvalue.html#D1">ABSVALUE:def 1</a></i>
<br/>.= 
<font color="Maroon">r</font> <a href="xcmplx_0.html#K3">*</a> 1
<b>by </b><i/>
;<br/>
<b>then reconsider </b><font color="Maroon">y</font> = <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">v</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span> <b>by </b><i>, <a class="ref" href="topreal9.html#T9">TOPREAL9:9</a></i>;<br/>
<b>take </b>
<font color="Maroon">y</font>
;<br/>

<b>thus </b><a NAME="E4:53_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">u</font>,<font color="Maroon">y</font>]
 ;<br/>


</div><b>end;</b></div>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span>,<span class="p1">(<span class="default"><a href="toprealb.html#K7" target="_self">Tcircle</a> <font color="Maroon">x</font>,<font color="Maroon">r</font></span>)</span><b> such that </b><br/><a NAME="E5:53"/><i><font color="Green">E69</font></i>: 
for <font color="Olive">x</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span> holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i>
<b>take </b>
<font color="Maroon">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="t_0topsp.html#D1">T_0TOPSP:def 1</a><br/>

<a NAME="E6:53"/>
for <font color="Olive">a</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span><br/> for <font color="Olive">b</font> being  <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>  st <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <font color="Olive">b</font> holds <br/><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Olive">b</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="53_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="toprealb.html#K8" target="_self">Tunit_circle</a> <font color="Maroon">n</font></span>)</span>;<br/><b>let </b><font color="Maroon">b</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/>



<a NAME="E1:53_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">a</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font>]
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E2:53_2"/>
( <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b</font> implies <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">r</font> <a href="euclid.html#K18">*</a> <font color="Maroon">b</font></span>)</span> <a href="euclid.html#K17">+</a> <font color="Maroon">x</font> )
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:53"/>
<font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i/>;<br/>


</div>
