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<b>let </b><font color="Maroon">GX</font> be  non <a href="struct_0.html#V3">empty</a> <a href="pre_topc.html#NM1">TopSpace</a>;<br/><b>let </b><font color="Maroon">A</font> be   <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font>;<br/>



<b>assume </b><a NAME="E1:4"/><i><font color="Green">E28</font></i>: 
for <font color="Olive">xa</font>, <font color="Olive">ya</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>  st <font color="Olive">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">ya</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">xa</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Olive">ya</font> holds <br/> ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">xa</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">ya</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 )
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1"/>
(  not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> or <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1"><b>suppose </b></a><a NAME="E1:4_1_1"/>
not <font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">A</font> = <font color="Maroon">A</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/><a NAME="E3:4_1_1"/><i><font color="Green">E30</font></i>: 
for <font color="Olive">xa</font>, <font color="Olive">ya</font> being  <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>  st <font color="Olive">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">ya</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Olive">xa</font> <a href="hidden.html#R1">=</a> <font color="Olive">ya</font> holds <br/> ex <font color="Olive">h</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">xa</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">ya</font> <a href="hidden.html#R1">=</a> <font color="Olive">h</font> <a href="funct_1.html#K1">.</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">xa</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>, <font color="Maroon">ya</font> be   <a href="pre_topc.html#NM2">Point</a> of <font color="Maroon">GX</font>;<br/>

<b>assume </b><a NAME="E1:4_1_1_1"/><i><font color="Green">E31</font></i>: 
( <font color="Maroon">xa</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">ya</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">ya</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">xa'</font> = <font color="Maroon">xa</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <b>by </b><i/>;<br/>
<b>reconsider </b><font color="Maroon">h</font> = <a href="borsuk_1.html#K20">I[01]</a>  <a href="borsuk_1.html#K3">--&gt;</a> <font color="Maroon">xa'</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon">h</font>
;<br/>

<a NAME="E4:4_1_1_1"/>
<font color="Maroon">h</font> <a href="funct_2.html#R2">=</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  <a href="borsuk_1.html#K1">--&gt;</a> <font color="Maroon">xa'</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="E5:4_1_1_1"/>
( <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">xa</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">ya</font> <a href="hidden.html#R1">=</a> <font color="Maroon">h</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, , <a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>;<br/>


</div><b>end;</b></div><a NAME="E4:4_1_1"/>
for <font color="Olive">xb</font>, <font color="Olive">yb</font> being  <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>  ex <font color="Olive">ha</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">ha</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Olive">xb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Olive">yb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 1 )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="4_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">xb</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">yb</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>;<br/>

<a NAME="E1:4_1_1_2"/>
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> &amp; <font color="Maroon">yb</font> <a href="hidden.html#R2">in</a>  <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> )
 
;<br/>
<a NAME="E2:4_1_1_2"/><b>then </b><i><font color="Green">E32</font></i>: 
( <font color="Maroon">xb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> &amp; <font color="Maroon">yb</font> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:4_1_1_2_1"/>
( <font color="Maroon">xb</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">yb</font> or <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yb</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2_1_1"><b>suppose </b></a><a NAME="E1:4_1_1_2_1_1"/>
<font color="Maroon">xb</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> <font color="Maroon">yb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_2_1_1"/>
 ex <font color="Olive">ha</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">ha</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">yb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_1_2_1_2"><b>suppose </b></a><a NAME="E1:4_1_1_2_1_2"/>
<font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Maroon">yb</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:4_1_1_2_1_2"/>
 ex <font color="Olive">ha</font> being  <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> st <br/>( <font color="Olive">ha</font> is <a href="pre_topc.html#V5">continuous</a> &amp; <font color="Maroon">xb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 0 &amp; <font color="Maroon">yb</font> <a href="hidden.html#R1">=</a> <font color="Olive">ha</font> <a href="funct_1.html#K1">.</a> 1 )
 <b>by </b><i>, </i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><a NAME="E5:4_1_1"/><b>then </b>
<font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font> is <a href="connsp_1.html#V1">connected</a>
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E6:4_1_1"/>
<font color="Maroon">A</font> is <a href="connsp_1.html#V2">connected</a>
 <b>by </b><i><a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="4_1_2"><b>suppose </b></a><a NAME="E1:4_1_2"/>
<font color="Maroon">A</font> is <a href="xboole_0.html#V1">empty</a>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">D</font> = <font color="Maroon">A</font> as  <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <font color="Maroon">GX</font> ;<br/><b>let </b><font color="Maroon">A1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>, <font color="Maroon">B1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span>; <i><font color="Red">:: according to </font></i><a class="ref" href="connsp_1.html#D2">CONNSP_1:def 2</a>,<a class="ref" href="connsp_1.html#D3">CONNSP_1:def 3</a><br/><b>assume </b><b>that </b><br/><a NAME="E3:4_1_2"/><i><font color="Green">E33</font></i>: 
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A1</font> <a href="subset_1.html#K4">\/</a> <font color="Maroon">B1</font>
 <b>and </b><br/><a NAME="E4:4_1_2"/>
<font color="Maroon">A1</font>,<font color="Maroon">B1</font> <a href="connsp_1.html#R1">are_separated</a> 
 ;<br/><a NAME="E5:4_1_2"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">D</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">D</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><b>hence </b><a NAME="E6:4_1_2"/>
( <font color="Maroon">A1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> or <font color="Maroon">B1</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K1">{}</a> <span class="p1">(<span class="default"><font color="Maroon">GX</font> <a href="pre_topc.html#K3">|</a> <font color="Maroon">A</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T15">XBOOLE_1:15</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
