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

<b>let </b><font color="Maroon" title="c1">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">p1</font>, <font color="Maroon" title="c3">p2</font>, <font color="Maroon" title="c4">q1</font>, <font color="Maroon" title="c5">q2</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> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:20"/><i><font color="Green" title="E18">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c4">q1</font>,<font color="Maroon" title="c5">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c2">p1</font>,<font color="Maroon" title="c3">p2</font> )
 ;<br/>

<b>hence </b><a NAME="E2:20"/>
( <font color="Maroon" title="c5">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> )
 <b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a><br/>

<b>reconsider </b><font color="Maroon" title="c6">P'</font> = <font color="Maroon" title="c1">P</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>let </b><font color="Maroon" title="c7">f</font> be   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">s1</font>, <font color="Maroon" title="c9">s2</font> be   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a>;<br/>



<b>assume </b><a NAME="E4:20"/><i><font color="Green" title="E19">A2</font></i>: 
( <font color="Maroon" title="c7">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font> &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s1</font> &amp; <font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c7">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s2</font> &amp; <font color="Maroon" title="c9">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 ;<br/>

<a NAME="E5:20"/><b>then </b><i><font color="Green" title="E20">A3</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E6:20"/><i><font color="Green" title="E21">A4</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 0 <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E7:20"/><i><font color="Green" title="E22">A5</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<a NAME="E8:20"/><i><font color="Green" title="E23">A6</font></i>: 
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="real_1.html#T92" title="REAL_1:th.92">REAL_1:92</a></i>;<br/>
<b>set </b><font color="Maroon" title="c10">Ex</font> =  <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span>;<br/>
<a NAME="E9:20"/><i><font color="Green" title="E24">A7</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="treal_1.html#T21" title="TREAL_1:th.21">TREAL_1:21</a></i>;<br/>
<b>set </b><font color="Maroon" title="c11">g</font> = <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span>;<br/>
<a NAME="E10:20"/><i><font color="Green" title="E25">A8</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12" title="TREAL_1:th.12">TREAL_1:12</a></i>;<br/>
<a NAME="E11:20"/><i><font color="Green" title="E26">A9</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12" title="TREAL_1:th.12">TREAL_1:12</a></i>;<br/>
<a NAME="E12:20"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E13:20"/><b>then </b><i><font color="Green" title="E27">A10</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E9:20"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E14:20"/><b>then </b><i><font color="Green" title="E28">A11</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>;<br/>
<a NAME="E15:20"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c7">f</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E13:20"><i><font color="Green" title="E27">A10</font></i></a>, <a class="ref" href="relat_1.html#T47" title="RELAT_1:th.47">RELAT_1:47</a></i>;<br/>
<a NAME="E16:20"/><b>then </b><i><font color="Green" title="E29">A12</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E17:20"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E9:20"><i><font color="Green" title="E24">A7</font></i></a>, <a class="txt" href="jordan6.html#E14:20"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E18:20"/><b>then </b><i><font color="Green" title="E30">A13</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E16:20"><i><font color="Green" title="E29">A12</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c7">f</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P'</font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E20:20"/><i><font color="Green" title="E31">A14</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="jordan6.html#E9:20"><i><font color="Green" title="E24">A7</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71" title="TOPS_2:th.71">TOPS_2:71</a></i>;<br/>
<a NAME="E21:20"/><i><font color="Green" title="E32">A15</font></i>: 
( 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">g</font> &amp; 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c12">g</font> )
 
<b>by </b><i><a class="txt" href="jordan6.html#E5:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan6.html#E6:20"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="jordan6.html#E7:20"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="jordan6.html#E8:20"><i><font color="Green" title="E23">A6</font></i></a>, <a class="txt" href="jordan6.html#E18:20"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a></i>;<br/>
<a NAME="E22:20"/><i><font color="Green" title="E33">A16</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">p1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="jordan6.html#E11:20"><i><font color="Green" title="E26">A9</font></i></a>, <a class="txt" href="jordan6.html#E18:20"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>;<br/>
<a NAME="E23:20"/><i><font color="Green" title="E34">A17</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="jordan6.html#E10:20"><i><font color="Green" title="E25">A8</font></i></a>, <a class="txt" href="jordan6.html#E18:20"><i><font color="Green" title="E30">A13</font></i></a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">qs1</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font>, <font color="Maroon" title="c14">qs2</font> = 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="jordan6.html#E5:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan6.html#E6:20"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="jordan6.html#E7:20"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="jordan6.html#E8:20"><i><font color="Green" title="E23">A6</font></i></a>, <a class="ref" href="borsuk_1.html#T86" title="BORSUK_1:th.86">BORSUK_1:86</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<i><font color="Green" title="E35">A18</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c13">qs1</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3" title="TREAL_1:def.3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c8">s1</font>

;<br/>
<i><font color="Green" title="E36">A19</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c14">qs2</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font></span>)</span></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 1</span>)</span> <a href="real_1.html#K3" title="REAL_1:func.3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font></span>)</span> <a href="real_1.html#K4" title="REAL_1:func.4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8" title="TREAL_1:th.8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3" title="TREAL_1:def.3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon" title="c9">s2</font>

;<br/>
<a NAME="E27:20"/><i><font color="Green" title="E37">A20</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q2</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="jordan6.html#E21:20"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="jordan6.html#E25:20"><i><font color="Green" title="E35">A18</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E28:20"/>
<font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">q1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:20"><i><font color="Green" title="E19">A2</font></i></a>, <a class="txt" href="jordan6.html#E21:20"><i><font color="Green" title="E32">A15</font></i></a>, <a class="txt" href="jordan6.html#E26:20"><i><font color="Green" title="E36">A19</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/>
<a NAME="E29:20"/><b>then </b>
1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c8">s1</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:20"><i><font color="Green" title="E18">A1</font></i></a>, <a class="txt" href="jordan6.html#E5:20"><i><font color="Green" title="E20">A3</font></i></a>, <a class="txt" href="jordan6.html#E6:20"><i><font color="Green" title="E21">A4</font></i></a>, <a class="txt" href="jordan6.html#E7:20"><i><font color="Green" title="E22">A5</font></i></a>, <a class="txt" href="jordan6.html#E20:20"><i><font color="Green" title="E31">A14</font></i></a>, <a class="txt" href="jordan6.html#E22:20"><i><font color="Green" title="E33">A16</font></i></a>, <a class="txt" href="jordan6.html#E23:20"><i><font color="Green" title="E34">A17</font></i></a>, <a class="txt" href="jordan6.html#E27:20"><i><font color="Green" title="E37">A20</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E30:20"/><b>then </b>
<font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5" title="REAL_1:func.5">-</a> <font color="Maroon" title="c9">s2</font></span>)</span>
 
<b>by </b><i><a class="ref" href="real_2.html#T165" title="REAL_2:th.165">REAL_2:165</a></i>;<br/>
<b>hence </b><a NAME="E31:20"/>
<font color="Maroon" title="c8">s1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c9">s2</font>
 ;<br/>


</div>
