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

<b>let </b><font color="Maroon" title="c7">P</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</font>, <font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c11">q2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E1:70"/><i><font color="Green" title="E49">A1</font></i>: 
<font color="Maroon" title="c7">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <b>and </b><br/><a NAME="E2:70"/><i><font color="Green" title="E50">A2</font></i>: 
( <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c11">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font> )
 <b>and </b><br/><a NAME="E3:70"/><i><font color="Green" title="E51">A3</font></i>: 
( <font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p1</font> &amp; <font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c11">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c8">p1</font> &amp; <font color="Maroon" title="c11">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c9">p2</font> )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:70_1"/>
( <font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q2</font> or <font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_1"><b>suppose </b></a><a NAME="E1:70_1_1"/>
<font color="Maroon" title="c10">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">q2</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c12">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="c10">q1</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font> <b>by </b><i><a class="ref" href="jordan.html#T41" target="_self" title="JORDAN:th.41">Th41</a></i>;<br/><b>take </b>
<font color="Maroon" title="c12">f</font>
;<br/><a NAME="E3:70_1_1"/><i><font color="Green" title="E52">A4</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> <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"><font color="Maroon" title="c10">q1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="funcop_1.html#T14" title="FUNCOP_1:th.14">FUNCOP_1:14</a></i>;<br/><b>thus </b><a NAME="E4:70_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:70_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font>
 ;<br/>

<b>hence </b><a NAME="E2:70_1_1_1"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">P</font>
 <b>by </b><i><a class="txt" href="jordan.html#E2:70"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="jordan.html#E3:70_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>


</div><b>end;</b></div><a NAME="E5:70_1_1"/>
( not <font color="Maroon" title="c8">p1</font> <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"><font color="Maroon" title="c10">q1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> &amp; not <font color="Maroon" title="c9">p2</font> <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"><font color="Maroon" title="c10">q1</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 <b>by </b><i><a class="txt" href="jordan.html#E3:70"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><b>hence </b><a NAME="E6:70_1_1"/>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c12">f</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 <b>by </b><i><a class="txt" href="jordan.html#E3:70_1_1"><i><font color="Green" title="E52">A4</font></i></a>, <a class="ref" href="zfmisc_1.html#T57" title="ZFMISC_1:th.57">ZFMISC_1:57</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="70_1_2"><b>suppose </b></a><a NAME="E1:70_1_2"/>
<font color="Maroon" title="c10">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c11">q2</font>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c12">Q</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:70_1_2"/><i><font color="Green" title="E52">A5</font></i>: 
<font color="Maroon" title="c12">Q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font>
 <b>and </b><br/><a NAME="E4:70_1_2"/><i><font color="Green" title="E53">A6</font></i>: 
( <font color="Maroon" title="c12">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> &amp; <font color="Maroon" title="c12">Q</font> <a href="subset_1.html#R1" title="SUBSET_1:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="txt" href="jordan.html#E1:70"><i><font color="Green" title="E49">A1</font></i></a>, <a class="txt" href="jordan.html#E2:70"><i><font color="Green" title="E50">A2</font></i></a>, <a class="txt" href="jordan.html#E3:70"><i><font color="Green" title="E51">A3</font></i></a>, <a class="ref" href="jordan16.html#T38" title="JORDAN16:th.38">JORDAN16:38</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">g</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font>, <font color="Maroon" title="c14">f</font> being   <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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">Q</font></span>)</span><b> such that </b><br/><a NAME="E6:70_1_2"/><i><font color="Green" title="E54">A7</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c14">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q</font> &amp; <font color="Maroon" title="c13">g</font> <a href="funct_2.html#R1" title="FUNCT_2:pred.1">=</a> <font color="Maroon" title="c14">f</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E3:70_1_2"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="jordan.html#T42" target="_self" title="JORDAN:th.42">Th42</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c15">h</font> = <font color="Maroon" title="c14">f</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="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreala.html#T28" title="TOPREALA:th.28">TOPREALA:28</a></i>;<br/><a NAME="E8:70_1_2"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">Q</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">Q</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c16">z1</font> = <font color="Maroon" title="c10">q1</font>, <font color="Maroon" title="c17">z2</font> = <font color="Maroon" title="c11">q2</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c12">Q</font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E3:70_1_2"><i><font color="Green" title="E52">A5</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><a NAME="E10:70_1_2"/><i><font color="Green" title="E55">A8</font></i>: 
<font color="Maroon" title="c16">z1</font>,<font color="Maroon" title="c17">z2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_1"><b>proof </b></a><div class="add">

<b>take </b>
<font color="Maroon" title="c14">f</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a><br/>

<b>thus </b><a NAME="E1:70_1_2_1"/>
<font color="Maroon" title="c14">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E6:70_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/>

<b>thus </b><a NAME="E2:70_1_2_1"/>
( <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z1</font> &amp; <font color="Maroon" title="c14">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="c17">z2</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E6:70_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E11:70_1_2"/>
<font color="Maroon" title="c14">f</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c16">z1</font>,<font color="Maroon" title="c17">z2</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="70_1_2_2"><b>proof </b></a><div class="add">

<b>thus </b><a NAME="E1:70_1_2_2"/>
<font color="Maroon" title="c16">z1</font>,<font color="Maroon" title="c17">z2</font> <a href="borsuk_6.html#R1" title="BORSUK_6:pred.1">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan.html#E10:70_1_2"><i><font color="Green" title="E55">A8</font></i></a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a><br/>

<b>thus </b><a NAME="E2:70_1_2_2"/>
<font color="Maroon" title="c14">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan.html#E6:70_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="topmetr.html#T8" title="TOPMETR:th.8">TOPMETR:8</a></i>;<br/>

<b>thus </b><a NAME="E3:70_1_2_2"/>
( <font color="Maroon" title="c14">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">z1</font> &amp; <font color="Maroon" title="c14">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="c17">z2</font> )
 <b>by </b><i><a class="txt" href="jordan.html#E6:70_1_2"><i><font color="Green" title="E54">A7</font></i></a>, <a class="ref" href="borsuk_2.html#D4" title="BORSUK_2:def.4">BORSUK_2:def 4</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c18">h</font> = <font color="Maroon" title="c15">h</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c10">q1</font>,<font color="Maroon" title="c11">q2</font> <b>by </b><i><a class="txt" href="jordan.html#E10:70_1_2"><i><font color="Green" title="E55">A8</font></i></a>, <a class="ref" href="topalg_2.html#T1" title="TOPALG_2:th.1">TOPALG_2:1</a></i>;<br/><b>take </b>
<font color="Maroon" title="c18">h</font>
;<br/><b>thus </b><a NAME="E13:70_1_2"/>
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">h</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">P</font> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">h</font> <a href="xboole_0.html#R1" title="XBOOLE_0:pred.1">misses</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> )
 <b>by </b><i><a class="txt" href="jordan.html#E4:70_1_2"><i><font color="Green" title="E53">A6</font></i></a>, <a class="txt" href="jordan.html#E6:70_1_2"><i><font color="Green" title="E54">A7</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
