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

<b>let </b><font color="Maroon">c<sub>1</sub></font>, <font color="Maroon">c<sub>2</sub></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> 2</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>3</sub></font>, <font color="Maroon">c<sub>4</sub></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> 2</span>)</span>;<br/>



<b>assume </b><a NAME="E1:22"/><i><font color="Green">E18</font></i>: 
( <font color="Maroon">c<sub>1</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a>  &amp; <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> is <a href="pre_topc.html#V4">closed</a> )
 ; <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a><br/>

<b>then consider </b><font color="Maroon">c<sub>5</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:22"/><i><font color="Green">E19</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>1</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>6</sub></font> = <font color="Maroon">c<sub>1</sub></font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan5c.html#E3:22"><i><font color="Green">E19</font></i></a></i>;<br/>
<a NAME="E5:22"/><i><font color="Green">E20</font></i>: 
( <font color="Maroon">c<sub>6</sub></font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>6</sub></font> <a href="xboole_0.html#NR5" target="_self">meets</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan5b.html#T14">JORDAN5B:14</a>, <a class="ref" href="xboole_0.html#D7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E6:22"/><b>then </b><i><font color="Green">E21</font></i>: 
 <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="ref" href="jordan5c.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E7:22"/><i><font color="Green">E22</font></i>: 
(  <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>6</sub></font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5c.html#E5:22"><i><font color="Green">E20</font></i></a>, <a class="ref" href="jordan5c.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan5c.html#D2" target="_self">Def2</a></i>;<br/>
<a NAME="E8:22"/><i><font color="Green">E23</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:22_1"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>;<br/>
<a NAME="E2:22_1"/><i><font color="Green">E25</font></i>: 
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="ref" href="treal_1.html#T21">TREAL_1:21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>;<br/>
<a NAME="E3:22_1"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E4:22_1"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E5:22_1"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E6:22_1"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:22_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:22_1"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:22_1"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E6:22_1"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E9:22_1"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:22_1"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:22_1"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan5c.html#E7:22_1"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:22_1"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E9:22_1"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> as   <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E13:22_1"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E2:22_1"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E14:22_1"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E4:22_1"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan5c.html#E11:22_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<a NAME="E15:22_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E3:22_1"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan5c.html#E11:22_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E16:22_1"/><i><font color="Green">E35</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E17:22_1"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E18:22_1"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E19:22_1"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E16:22_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan5c.html#E17:22_1"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="jordan5c.html#E18:22_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>

;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>

;<br/>
<a NAME="E23:22_1"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E11:22_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan5c.html#E18:22_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E24:22_1"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E21:22_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E25:22_1"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E11:22_1"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan5c.html#E19:22_1"><i><font color="Green">E38</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E26:22_1"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E22:22_1"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E27:22_1"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E16:22_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E28:22_1"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5c.html#E5:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5c.html#E1:22_1"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E13:22_1"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan5c.html#E14:22_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan5c.html#E15:22_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan5c.html#E18:22_1"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan5c.html#E19:22_1"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan5c.html#E24:22_1"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan5c.html#E26:22_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan5c.html#D2" target="_self">Def2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:22"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span><br/>for <font color="Olive">b<sub>2</sub></font> being  <a href="real_1.html#NM1">Real</a> st <font color="Olive">b<sub>1</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">b<sub>2</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 holds <br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="real_1.html#NM1">Real</a> st 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Olive">b<sub>3</sub></font> &amp; <font color="Olive">b<sub>3</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Olive">b<sub>2</sub></font> holds <br/> not <font color="Olive">b<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="22_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>7</sub></font> be   <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>8</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>



<b>assume </b><a NAME="E1:22_2"/><i><font color="Green">E24</font></i>: 
( <font color="Maroon">c<sub>7</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">c<sub>6</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp; 0 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 ;<br/>

<b>set </b><font color="Maroon">c<sub>9</sub></font> =  <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span>;<br/>
<a NAME="E2:22_2"/><i><font color="Green">E25</font></i>: 
 <a href="treal_1.html#K3">L[01]</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="ref" href="treal_1.html#T21">TREAL_1:21</a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>10</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>;<br/>
<a NAME="E3:22_2"/><i><font color="Green">E26</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E4:22_2"/><i><font color="Green">E27</font></i>: 
<span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#T12">TREAL_1:12</a></i>;<br/>
<a NAME="E5:22_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E6:22_2"/><b>then </b><i><font color="Green">E28</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:22_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E7:22_2"/><b>then </b><i><font color="Green">E29</font></i>: 
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<a NAME="E8:22_2"/>
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>7</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E6:22_2"><i><font color="Green">E28</font></i></a>, <a class="ref" href="relat_1.html#T47">RELAT_1:47</a></i>;<br/>
<a NAME="E9:22_2"/><b>then </b><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:22_2"/>
 <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K5">I[01]</a> 
 
<b>by </b><i><a class="txt" href="jordan5c.html#E2:22_2"><i><font color="Green">E25</font></i></a>, <a class="txt" href="jordan5c.html#E7:22_2"><i><font color="Green">E29</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:22_2"/><b>then </b><i><font color="Green">E31</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p3">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p3">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span></span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E9:22_2"><i><font color="Green">E30</font></i></a>, <a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>11</sub></font> = <font color="Maroon">c<sub>7</sub></font> <a href="partfun1.html#K1">*</a> <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> as   <a href="struct_0.html#NM7">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> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">c<sub>6</sub></font></span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E13:22_2"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>11</sub></font> is <a href="tops_2.html#V3">being_homeomorphism</a>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E2:22_2"><i><font color="Green">E25</font></i></a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#T71">TOPS_2:71</a></i>;<br/>
<a NAME="E14:22_2"/><i><font color="Green">E33</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E4:22_2"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan5c.html#E11:22_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<a NAME="E15:22_2"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E3:22_2"><i><font color="Green">E26</font></i></a>, <a class="txt" href="jordan5c.html#E11:22_2"><i><font color="Green">E31</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a></i>;<br/>
<b>let </b><font color="Maroon">c<sub>12</sub></font> be   <a href="real_1.html#NM1">Real</a>;<br/>

<b>assume </b><a NAME="E16:22_2"/><i><font color="Green">E35</font></i>: 
( 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">c<sub>12</sub></font> &amp; <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">c<sub>8</sub></font> )
 ;<br/>

<a NAME="E17:22_2"/><b>then </b><i><font color="Green">E36</font></i>: 
<font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a></i>;<br/>
<b>set </b><font color="Maroon">c<sub>13</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font>;<br/>
<a NAME="E18:22_2"/><i><font color="Green">E37</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<a NAME="E19:22_2"/><i><font color="Green">E38</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;=</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> &amp; 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="xxreal_0.html#R1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan5c.html#E16:22_2"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan5c.html#E17:22_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan5c.html#E1"><i><font color="Green">Lemma1</font></i></a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>14</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font>, <font color="Maroon">c<sub>15</sub></font> = 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span> <b>by </b><i><a class="txt" href="jordan5c.html#E18:22_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<i><font color="Green">E39</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>14</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font>

;<br/>
<i><font color="Green">E40</font></i>: <span class="p1">(<span class="default"><a href="treal_1.html#K3">L[01]</a> <span class="p2">(<span class="default">0,1 <a href="treal_1.html#K2">(#)</a> </span>)</span>,<span class="p2">(<span class="default"><a href="treal_1.html#K1">(#)</a> 0,1</span>)</span></span>)</span> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>15</sub></font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <span class="p3">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span></span>)</span> <a href="real_1.html#K4">*</a> 1</span>)</span> <a href="real_1.html#K3">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font></span>)</span> <a href="real_1.html#K4">*</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="treal_1.html#T8">TREAL_1:8</a>, <a class="ref" href="treal_1.html#D3">TREAL_1:def 3</a></i>
<br/>.= 
<font color="Maroon">c<sub>12</sub></font>

;<br/>
<a NAME="E23:22_2"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E11:22_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan5c.html#E18:22_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E24:22_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default">1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>8</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E21:22_2"><i><font color="Green">E39</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E25:22_2"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>11</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E11:22_2"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan5c.html#E19:22_2"><i><font color="Green">E38</font></i></a>, <a class="ref" href="borsuk_1.html#T86">BORSUK_1:86</a></i>;<br/>
<a NAME="E26:22_2"/><b>then </b><i><font color="Green">E42</font></i>: 
<font color="Maroon">c<sub>11</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>15</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E22:22_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/>
<a NAME="E27:22_2"/>
1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>8</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 1 <a href="real_1.html#K5">-</a> <font color="Maroon">c<sub>12</sub></font>
 
<b>by </b><i><a class="txt" href="jordan5c.html#E16:22_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="real_1.html#T92">REAL_1:92</a></i>;<br/>
<b>hence </b><a NAME="E28:22_2"/>
not <font color="Maroon">c<sub>7</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>12</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>2</sub></font>
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5c.html#E5:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5c.html#E1:22_2"><i><font color="Green">E24</font></i></a>, <a class="txt" href="jordan5c.html#E13:22_2"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan5c.html#E14:22_2"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan5c.html#E15:22_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan5c.html#E18:22_2"><i><font color="Green">E37</font></i></a>, <a class="txt" href="jordan5c.html#E19:22_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan5c.html#E24:22_2"><i><font color="Green">E41</font></i></a>, <a class="txt" href="jordan5c.html#E26:22_2"><i><font color="Green">E42</font></i></a>, <a class="ref" href="jordan5c.html#D1" target="_self">Def1</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:22"/>
(  <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> &amp;  <a href="jordan5c.html#K2" target="_self">Last_Point</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan5c.html#K1" target="_self">First_Point</a> <font color="Maroon">c<sub>1</sub></font>,<font color="Maroon">c<sub>4</sub></font>,<font color="Maroon">c<sub>3</sub></font>,<font color="Maroon">c<sub>2</sub></font> )
 <b>by </b><i><a class="txt" href="jordan5c.html#E1:22"><i><font color="Green">E18</font></i></a>, <a class="txt" href="jordan5c.html#E5:22"><i><font color="Green">E20</font></i></a>, <a class="txt" href="jordan5c.html#E6:22"><i><font color="Green">E21</font></i></a>, <a class="txt" href="jordan5c.html#E7:22"><i><font color="Green">E22</font></i></a>, <a class="txt" href="jordan5c.html#E8:22"><i><font color="Green">E23</font></i></a>, <a class="ref" href="jordan5c.html#D1" target="_self">Def1</a>, <a class="ref" href="jordan5c.html#D2" target="_self">Def2</a></i>;<br/>


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