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

<b>set </b><font color="Maroon">c<sub>1</sub></font> = <a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span>;<br/>
<a NAME="E1:14_1_1"/><i><font color="Green">E6</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E2:14_1_1"/>
(  <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  &amp;  <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R2">in</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">c<sub>2</sub></font> =  <a href="borsuk_1.html#K21">0[01]</a>  as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:14_1_1"/>
for <font color="Olive">b<sub>1</sub></font>, <font color="Olive">b<sub>2</sub></font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span></span>)</span> holds <font color="Olive">b<sub>1</sub></font>,<font color="Olive">b<sub>2</sub></font> <a href="borsuk_2.html#R2" target="_self">are_connected</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1_1_1"><b>proof </b></a><div class="add">

<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="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span></span>)</span>;<br/>

<a NAME="E1:14_1_1_1"/><i><font color="Green">E7</font></i>: 
( <font color="Maroon">c<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> &amp; <font color="Maroon">c<sub>4</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>2</sub></font> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:14_1_1"><i><font color="Green">E6</font></i></a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon">c<sub>5</sub></font> = <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">c<sub>2</sub></font> as   <a href="struct_0.html#NM7">Function</a> of <a href="topmetr.html#K5">I[01]</a> ,<span class="p1">(<span class="default"><a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p2"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">c<sub>5</sub></font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" target="_self">BORSUK_2:def 1</a><br/>

<b>thus </b><a NAME="E3:14_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> is <a href="pre_topc.html#V5">continuous</a>
 ;<br/>

<a NAME="E4:14_1_1_1"/>
<font color="Maroon">c<sub>5</sub></font> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="topmetr.html#K5">I[01]</a>  <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">c<sub>2</sub></font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#D2">BORSUK_1:def 2</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1_1_1"/>
( <font color="Maroon">c<sub>5</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>5</sub></font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">c<sub>4</sub></font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E1:14_1_1_1"><i><font color="Green">E7</font></i></a>, <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="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:14_1_1"/><b>then </b>
<a href="topmetr.html#K5">I[01]</a>  <a href="pre_topc.html#K3">|</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><a href="borsuk_1.html#K21">0[01]</a> </span><a href="domain_1.html#K6">}</a></span> is <a href="borsuk_2.html#V1" target="_self">arcwise_connected</a>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3" target="_self">Def3</a></i>;<br/>
<b>hence </b><a NAME="E6:14_1_1"/>
ex <font color="Olive">b<sub>1</sub></font> being  <a href="pre_topc.html#NM1">TopSpace</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="borsuk_2.html#V1" target="_self">arcwise_connected</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="struct_0.html#V3">empty</a> )
 ;<br/>


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