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

<b>let </b><font color="Maroon">c<sub>1</sub></font> be   <a href="struct_0.html#NM7">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>,<a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:34"/><i><font color="Green">E27</font></i>: 
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> holds <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Olive">b<sub>1</sub></font>,1
 ;<br/>

<a NAME="E2:34"/>
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K2">Seg</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E3:34"/><b>then </b><i><font color="Green">E28</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a>, <a class="ref" href="jordan2b.html#T22" target="_self">Th22</a></i>;<br/>
<a NAME="E4:34"/><i><font color="Green">E29</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<a NAME="E5:34"/><i><font color="Green">E30</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:34"/>
 <a href="numbers.html#K1">REAL</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K1">REAL</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> ;<br/>
<a NAME="E3:34_1"/><i><font color="Green">E31</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <span class="p1"><a href="jordan2b.html#K2" target="_self">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="jordan2b.html#K2" target="_self">]|</a></span> = 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <span class="p1"><a href="jordan2b.html#K2" target="_self">|[</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="jordan2b.html#K2" target="_self">]|</a></span>,1
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E3:34_1"><i><font color="Green">E31</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E5:34_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E5:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<b>then </b><i><font color="Green">E31</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
<b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>
;<br/>
<a NAME="E8:34"/><i><font color="Green">E32</font></i>: 
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3"><b>proof </b></a><div class="add">

<a NAME="E1:34_3"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Olive">b<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Olive">b<sub>2</sub></font> holds <br/><font color="Olive">b<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">b<sub>2</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_3_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>3</sub></font> be    <a href="hidden.html#M1">set</a> ;<br/>



<b>assume </b><a NAME="E1:34_3_1"/><i><font color="Green">E33</font></i>: 
( <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>1</sub></font> &amp; <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>1</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>3</sub></font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>4</sub></font> = <font color="Maroon">c<sub>2</sub></font>, <font color="Maroon">c<sub>5</sub></font> = <font color="Maroon">c<sub>3</sub></font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> ;<br/>
<b>consider </b><font color="Maroon">c<sub>6</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E4:34_3_1"/><i><font color="Green">E34</font></i>: 
<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2b.html#T24" target="_self">Th24</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E6:34_3_1"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2b.html#T24" target="_self">Th24</a></i>;<br/>
<a NAME="E7:34_3_1"/><i><font color="Green">E36</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<a NAME="E8:34_3_1"/><i><font color="Green">E37</font></i>: 
( 1 <a href="xxreal_0.html#R1">&lt;=</a> 1 &amp; 1 <a href="xxreal_0.html#R1">&lt;=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> )
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<i><font color="Green">E38</font></i>: <font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,1
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E4:34_3_1"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>6</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E7:34_3_1"><i><font color="Green">E36</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>6</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>5</sub></font> = 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>5</sub></font>,1
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E6:34_3_1"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E8:34_3_1"><i><font color="Green">E37</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>7</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E11:34_3_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>3</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:34_3_1"><i><font color="Green">E33</font></i></a>, <a class="txt" href="jordan2b.html#E4:34_3_1"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan2b.html#E6:34_3_1"><i><font color="Green">E35</font></i></a>, <a class="txt" href="jordan2b.html#E9:34_3_1"><i><font color="Green">E38</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:34_3"/>
<font color="Maroon">c<sub>1</sub></font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E9:34"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#K2">/"</a>  is <a href="pre_topc.html#V5">continuous</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4"><b>proof </b></a><div class="add">

<a NAME="E1:34_4"/>
the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>2</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 ;<br/>

<b>then reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>4</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<font color="Maroon">c<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Maroon">c<sub>4</sub></font> = 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>4</sub></font>,1
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>3</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>2</sub></font>
<b>by </b><i><a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
;<br/>
<b>hence </b><a NAME="E5:34_4_1"/>
<font color="Maroon">c<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font>
 <b>by </b><i><a class="txt" href="jordan2b.html#E5:34"><i><font color="Green">E30</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_4"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E3:34_4"/><b>then </b><i><font color="Green">E33</font></i>: 
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">c<sub>1</sub></font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="topmetr.html#K3">R^1</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#T12">PRE_TOPC:12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>2</sub></font> = <font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#K2">/"</a>  as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K3">R^1</a> ,<span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>3</sub></font> = <font color="Maroon">c<sub>2</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> ;<br/>
<a NAME="E6:34_4"/>
for <font color="Olive">b<sub>1</sub></font> being <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> <br/>for <font color="Olive">b<sub>2</sub></font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span><br/>for <font color="Olive">b<sub>3</sub></font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span></span>)</span> st <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Olive">b<sub>3</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Olive">b<sub>2</sub></font>,<font color="Olive">b<sub>1</sub></font> holds <br/><font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Olive">b<sub>3</sub></font> is <a href="pre_topc.html#V3">open</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>4</sub></font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">c<sub>5</sub></font> be    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span>;<br/><b>let </b><font color="Maroon">c<sub>6</sub></font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="pcomps_1.html#K4">TopSpaceMetr</a> <span class="p2">(<span class="default"><a href="euclid.html#K14">Euclid</a> 1</span>)</span></span>)</span>;<br/>





<b>assume </b><a NAME="E1:34_4_2"/><i><font color="Green">E34</font></i>: 
( <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 &amp; <font color="Maroon">c<sub>6</sub></font> <a href="hidden.html#R1">=</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> )
 ;<br/>

<b>consider </b><font color="Maroon">c<sub>7</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:34_4_2"/><i><font color="Green">E35</font></i>: 
<font color="Maroon">c<sub>5</sub></font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>7</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="jordan2b.html#T25" target="_self">Th25</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>8</sub></font> = <font color="Maroon">c<sub>1</sub></font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>,the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K3">R^1</a>  ;<br/>
<a NAME="E5:34_4_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E6:34_4_2"/><b>then </b><i><font color="Green">E36</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> <a href="hidden.html#R1">=</a>  <a href="euclid.html#K1">REAL</a> 1
 
<b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E7:34_4_2"/><i><font color="Green">E37</font></i>: 
<font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K2">"</a> 
 
<b>by </b><i><a class="txt" href="jordan2b.html#E8:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan2b.html#E3:34_4"><i><font color="Green">E33</font></i></a>, <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E8:34_4_2"/><i><font color="Green">E38</font></i>: 
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34_4_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan2b.html#E3:34_4_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan2b.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E9:34_4_2"/>
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2_1"><b>proof </b></a><div class="add">

<a NAME="E1:34_4_2_1"/><i><font color="Green">E39</font></i>: 
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="tarski.html#R1">c=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_4_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:34_4_2_1_1"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">c<sub>8</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> &amp; <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <font color="Maroon">c<sub>10</sub></font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon">c<sub>11</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E5:34_4_2_1_1"/><i><font color="Green">E41</font></i>: 
( <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>11</sub></font> &amp; <font color="Maroon">c<sub>11</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="jordan2b.html#E8:34_4_2"><i><font color="Green">E38</font></i></a>, <a class="txt" href="jordan2b.html#E3:34_4_2_1_1"><i><font color="Green">E40</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>12</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E7:34_4_2_1_1"/><i><font color="Green">E42</font></i>: 
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R1">=</a> 1
 
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> = 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>12</sub></font>,1
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a>, <a class="txt" href="jordan2b.html#E3:34_4_2_1_1"><i><font color="Green">E40</font></i></a>, <a class="txt" href="jordan2b.html#E5:34_4_2_1_1"><i><font color="Green">E41</font></i></a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>11</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="funct_1.html#K1">.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E7:34_4_2_1_1"><i><font color="Green">E42</font></i></a>, <a class="ref" href="finseq_4.html#T24">FINSEQ_4:24</a></i>
<br/>.= 
<font color="Maroon">c<sub>11</sub></font>
<b>by </b><i><a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>
;<br/>
<b>hence </b><a NAME="E9:34_4_2_1_1"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2b.html#E5:34_4_2_1_1"><i><font color="Green">E41</font></i></a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:34_4_2_1"/>
<span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span>  <a href="tarski.html#R1">c=</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="34_4_2_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">c<sub>9</sub></font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:34_4_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon">c<sub>10</sub></font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:34_4_2_1_2"/><i><font color="Green">E40</font></i>: 
( <font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>10</sub></font> &amp; <font color="Maroon">c<sub>10</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> )
 ;<br/>
<b>reconsider </b><font color="Maroon">c<sub>11</sub></font> = <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 1</span>)</span> <b>by </b><i><a class="ref" href="euclid.html#T25">EUCLID:25</a></i>;<br/>
<a NAME="E5:34_4_2_1_2"/>
 <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <span class="p2"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Olive">b<sub>1</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34_4_2"><i><font color="Green">E34</font></i></a>, <a class="txt" href="jordan2b.html#E3:34_4_2"><i><font color="Green">E35</font></i></a>, <a class="ref" href="jordan2b.html#T33" target="_self">Th33</a></i>;<br/>
<a NAME="E6:34_4_2_1_2"/><b>then </b><i><font color="Green">E41</font></i>: 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="hidden.html#R2">in</a>  <a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font>
 
<b>by </b><i><a class="txt" href="jordan2b.html#E3:34_4_2_1_2"><i><font color="Green">E40</font></i></a></i>;<br/>
<font color="Maroon">c<sub>9</sub></font> = 
<span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span> <a href="finseq_4.html#K4">/.</a> 1
<b>by </b><i><a class="txt" href="jordan2b.html#E3:34_4_2_1_2"><i><font color="Green">E40</font></i></a>, <a class="ref" href="finseq_4.html#T25">FINSEQ_4:25</a></i>
<br/>.= 
 <a href="jordan2b.html#K1" target="_self">Proj</a> <font color="Maroon">c<sub>11</sub></font>,1
<b>by </b><i><a class="ref" href="jordan2b.html#D1" target="_self">Def1</a></i>
<br/>.= 
<font color="Maroon">c<sub>8</sub></font> <a href="funct_1.html#K1">.</a> <span class="p1"><a href="binarith.html#K13">&lt;*</a><span class="default"><font color="Maroon">c<sub>10</sub></font></span><a href="binarith.html#K13">*&gt;</a></span>
<b>by </b><i><a class="txt" href="jordan2b.html#E1:34"><i><font color="Green">E27</font></i></a></i>
;<br/>
<b>hence </b><a NAME="E8:34_4_2_1_2"/>
<font color="Maroon">c<sub>9</sub></font> <a href="hidden.html#R2">in</a> <font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan2b.html#E6:34_4_2"><i><font color="Green">E36</font></i></a>, <a class="txt" href="jordan2b.html#E6:34_4_2_1_2"><i><font color="Green">E41</font></i></a>, <a class="ref" href="funct_1.html#D12">FUNCT_1:def 12</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:34_4_2_1"/>
<font color="Maroon">c<sub>8</sub></font> <a href="relset_1.html#K10">.:</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:34_4_2_1"><i><font color="Green">E39</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E10:34_4_2"/><b>then </b>
<font color="Maroon">c<sub>3</sub></font> <a href="relset_1.html#K11">"</a> <span class="p1">(<span class="default"><a href="metric_1.html#K9">Ball</a> <font color="Maroon">c<sub>5</sub></font>,<font color="Maroon">c<sub>4</sub></font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">{<span class="default"> <font color="Olive">b<sub>1</sub></font> where B is   <a href="real_1.html#NM1">Real</a> : ( <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K6">-</a> <font color="Maroon">c<sub>4</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Olive">b<sub>1</sub></font> &amp; <font color="Olive">b<sub>1</sub></font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">c<sub>7</sub></font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">c<sub>4</sub></font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan2b.html#E8:34"><i><font color="Green">E32</font></i></a>, <a class="txt" href="jordan2b.html#E7:34_4_2"><i><font color="Green">E37</font></i></a>, <a class="ref" href="funct_1.html#T154">FUNCT_1:154</a></i>;<br/>
<b>hence </b><a NAME="E11:34_4_2"/>
<font color="Maroon">c<sub>2</sub></font> <a href="relset_1.html#K11">"</a> <font color="Maroon">c<sub>6</sub></font> is <a href="pre_topc.html#V3">open</a>
 <b>by </b><i><a class="txt" href="jordan2b.html#E1:34_4_2"><i><font color="Green">E34</font></i></a>, <a class="ref" href="jordan2b.html#T32" target="_self">Th32</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E7:34_4"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#K2">/"</a>  is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="jordan2b.html#T20" target="_self">Th20</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E10:34"/>
<font color="Maroon">c<sub>1</sub></font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="jordan2b.html#E3:34"><i><font color="Green">E28</font></i></a>, <a class="txt" href="jordan2b.html#E4:34"><i><font color="Green">E29</font></i></a>, <a class="txt" href="jordan2b.html#E5:34"><i><font color="Green">E30</font></i></a>, <a class="txt" href="jordan2b.html#E7:34"><i><font color="Green">E31</font></i></a>, <a class="txt" href="jordan2b.html#E8:34"><i><font color="Green">E32</font></i></a>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>


</div>
