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

<b>set </b><font color="Maroon" title="c1">T</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>;<br/>
<a NAME="E1:14_1_1"/><i><font color="Green" title="E6">A1</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E2:14_1_1"/>
(  <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c2">nl</font> =  <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E4:14_1_1"/>
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> holds  <font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">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" title="c3">a</font>, <font color="Maroon" title="c4">b</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span>;<br/>

<a NAME="E1:14_1_1_1"/><i><font color="Green" title="E7">A2</font></i>: 
( <font color="Maroon" title="c3">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">nl</font> &amp; <font color="Maroon" title="c4">b</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">nl</font> )
 
<b>by </b><i><a class="txt" href="borsuk_2.html#E1:14_1_1"><i><font color="Green" title="E6">A1</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c5">f</font> = <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="borsuk_1.html#K3" title="BORSUK_1:func.3">--&gt;</a> <font color="Maroon" title="c2">nl</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p2"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c5">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" target="_self" title="BORSUK_2:def.1">BORSUK_2:def 1</a><br/>

<b>thus </b><a NAME="E3:14_1_1_1"/>
<font color="Maroon" title="c5">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 ;<br/>

<b>thus </b><a NAME="E4:14_1_1_1"/>
( <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp; <font color="Maroon" title="c5">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">b</font> )
 <b>by </b><i><a class="txt" href="borsuk_2.html#E1:14_1_1_1"><i><font color="Green" title="E7">A2</font></i></a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">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" title="TOPMETR:func.5">I[01]</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a> </span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a>
 
<b>by </b><i><a class="ref" href="borsuk_2.html#D3" target="_self" title="BORSUK_2:def.3">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" title="PRE_TOPC:NM.1">TopSpace</a> st <br/>( <font color="Olive">b<sub>1</sub></font> is <a href="borsuk_2.html#V1" title="BORSUK_2:attr.1">arcwise_connected</a> &amp; not <font color="Olive">b<sub>1</sub></font> is <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> )
 ;<br/>


</div>
