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

<b>set </b><font color="Maroon" title="c2">f</font> = <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>
<b>thus </b><a NAME="E1:35_1_1"/>
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> is <a href="topreal9.html#V1" title="TOPREAL9:attr.1">homogeneous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="35_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">r</font> be  <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal9.html#D4" target="_self" title="TOPREAL9:def.4">TOPREAL9:def 4</a><br/><b>let </b><font color="Maroon" title="c4">x</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>



<a NAME="E1:35_1_1_1"/><i><font color="Green" title="E22">A1</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2" title="BORSUK_1:def.2">BORSUK_1:def 2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <font color="Maroon" title="c4">x</font></span>)</span> = 
 <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T32" title="EUCLID:th.32">EUCLID:32</a></i>
<br/>.= 
<font color="Maroon" title="c3">r</font> <a href="euclid.html#K18" title="EUCLID:func.18">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">x</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E1:35_1_1_1"><i><font color="Green" title="E22">A1</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><br/>


</div><b>end;</b></div>

<b>let </b><font color="Maroon" title="c3">x</font>, <font color="Maroon" title="c4">y</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="topreal9.html#D5" target="_self" title="TOPREAL9:def.5">TOPREAL9:def 5</a><br/>

<a NAME="E2:35_1_1"/><i><font color="Green" title="E22">A2</font></i>: 
<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2" title="BORSUK_1:def.2">BORSUK_1:def 2</a></i>;<br/>
<b>hence </b><span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p2">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">x</font> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <font color="Maroon" title="c4">y</font></span>)</span> = 
 <a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="ref" href="euclid.html#T31" title="EUCLID:th.31">EUCLID:31</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E2:35_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">x</font></span>)</span> <a href="euclid.html#K17" title="EUCLID:func.17">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="borsuk_1.html#K2" title="BORSUK_1:func.2">--&gt;</a> <span class="p3">(<span class="default"><a href="euclid.html#K16" title="EUCLID:func.16">0.REAL</a> <font color="Maroon" title="c1">n</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">y</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal9.html#E2:35_1_1"><i><font color="Green" title="E22">A2</font></i></a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><br/>


</div>
