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<div class="add">

<b>let </b><font color="Maroon" title="c1">P</font>, <font color="Maroon" title="c2">Q1</font> be  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>;<br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c6">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><a NAME="E1:26"/><i><font color="Green" title="E28">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">Q1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font> )
 ;<br/>

<a NAME="E2:26"/><b>then </b><i><font color="Green" title="E29">A2</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E3:26"/><i><font color="Green" title="E30">A3</font></i>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<a NAME="E4:26"/><i><font color="Green" title="E31">A4</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E5:26"/><i><font color="Green" title="E32">A5</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan6.html#T49" title="JORDAN6:th.49">JORDAN6:49</a></i>;<br/>
<a NAME="E6:26"/><b>then </b><i><font color="Green" title="E33">A6</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan16.html#T36" title="JORDAN16:th.36">JORDAN16:36</a></i>;<br/>
<div><i><font color="Green" title="E34">A7</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_1"/><i><font color="Green" title="E35">A8</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><a NAME="E2:26_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:26_1"><i><font color="Green" title="E35">A8</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<div><i><font color="Green" title="E35">A9</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_2"/><i><font color="Green" title="E36">A10</font></i>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ;<br/><a NAME="E2:26_2"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="jordan20.html#E1:26_2"><i><font color="Green" title="E36">A10</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<b>reconsider </b><font color="Maroon" title="c7">Q0</font> =  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> as  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>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan16.html#T22" title="JORDAN16:th.22">JORDAN16:22</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3"/>
not <font color="Maroon" title="c2">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Q0</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c8">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:26_3"/><i><font color="Green" title="E36">A11</font></i>: 
( <font color="Maroon" title="c8">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q1</font> &amp; not <font color="Maroon" title="c8">x8</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q0</font> )
 <b>by </b><i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/><a NAME="E4:26_3"/>
<font color="Maroon" title="c7">Q0</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="jordan16.html#T36" title="JORDAN16:th.36">JORDAN16:36</a></i>;<br/><a NAME="E5:26_3"/><b>then </b><i><font color="Green" title="E37">A12</font></i>: 
( <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q0</font> &amp; <font color="Maroon" title="c6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Q0</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">q</font> = <font color="Maroon" title="c8">x8</font> as   <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> <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a></i>;<br/><a NAME="E7:26_3"/><i><font color="Green" title="E38">A13</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="jordan16.html#T9" title="JORDAN16:th.9">JORDAN16:9</a></i>;<br/><a NAME="E8:26_3"/><i><font color="Green" title="E39">A14</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 ;<br/><a NAME="E9:26_3"/><i><font color="Green" title="E40">A15</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E10:26_3"/><i><font color="Green" title="E41">A16</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><a NAME="E11:26_3"/><b>then </b><i><font color="Green" title="E42">A17</font></i>: 
<span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan20.html#T23" target="_self" title="JORDAN20:th.23">Th23</a></i>;<br/><a NAME="E12:26_3"/><i><font color="Green" title="E43">A18</font></i>: 
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3"><i><font color="Green" title="E41">A16</font></i></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></i>;<br/><a NAME="E13:26_3"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q2</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span> = 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3"><i><font color="Green" title="E43">A18</font></i></a>, <a class="ref" href="jordan20.html#T23" target="_self" title="JORDAN20:th.23">Th23</a></i>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="ref" href="jordan20.html#T25" target="_self" title="JORDAN20:th.25">Th25</a></i>
;<br/><a NAME="E15:26_3"/><b>then </b><i><font color="Green" title="E44">A19</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c6">q2</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1"/>
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font> )
 </span><b>by </b><i><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3"><i><font color="Green" title="E42">A17</font></i></a>, <a class="txt" href="jordan20.html#E15:26_3"><i><font color="Green" title="E44">A19</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p3</font> where <font color="Olive" title="b1">p3</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">p3</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p3</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p3</font> being   <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><b> such that </b><br/><a NAME="E4:26_3_2_1_1"/><i><font color="Green" title="E45">A20</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p3</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/><a NAME="E5:26_3_2_1_1"/><i><font color="Green" title="E46">A21</font></i>: 
not <font color="Maroon" title="c6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:26_3_2_1_1"/><i><font color="Green" title="E47">A22</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">Qa</font> = <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  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"><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="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3_2_1_1"><i><font color="Green" title="E46">A21</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">Qa'</font> = <font color="Maroon" title="c11">Qa</font> as   <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/><div><i><font color="Green" title="E48">A23</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_1_1"/><i><font color="Green" title="E49">A24</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font>
 ;<br/><a NAME="E2:26_3_2_1_1_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1"><i><font color="Green" title="E45">A20</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E6:26"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_1_1"><i><font color="Green" title="E49">A24</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:26_3_2_1_1"/><i><font color="Green" title="E49">A25</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E11:26_3_2_1_1"/><i><font color="Green" title="E50">A26</font></i>: 
not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:26_3_2_1_1"/><i><font color="Green" title="E51">A27</font></i>: 
not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26"><i><font color="Green" title="E34">A7</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:26_3_2_1_1"/><i><font color="Green" title="E52">A28</font></i>: 
not <font color="Maroon" title="c6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_1"/>
not <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green" title="E38">A13</font></i></a>, <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="c13">p1'</font> = <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c15">q2'</font> = <font color="Maroon" title="c6">q2</font>, <font color="Maroon" title="c16">p2'</font> = <font color="Maroon" title="c4">p2</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1"><i><font color="Green" title="E49">A25</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1"><i><font color="Green" title="E50">A26</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1"><i><font color="Green" title="E51">A27</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1"><i><font color="Green" title="E52">A28</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_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="E1:26_3_2_1_1_2_1"/>
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1_2_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E53">A29</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E26"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">f1</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_1_2_1_1"/><i><font color="Green" title="E54">A30</font></i>: 
( <font color="Maroon" title="c17">f1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c17">f1</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">p1</font> &amp; <font color="Maroon" title="c17">f1</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="c9">q</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_1_2_1_1"/><i><font color="Green" title="E55">A31</font></i>: 
<font color="Maroon" title="c17">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_1_2_1_1"/><i><font color="Green" title="E56">A32</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E57">A33</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_1_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_2_1_1_2"/><i><font color="Green" title="E58">A34</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1"><i><font color="Green" title="E45">A20</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A34</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_1_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_1_2"><i><font color="Green" title="E58">A34</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_1_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E58">A35</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">g1</font> = <font color="Maroon" title="c17">f1</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_1"><i><font color="Green" title="E49">A25</font></i></a>, <a class="txt" href="jordan20.html#E6:26_3_2_1_1_2_1_1"><i><font color="Green" title="E56">A32</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_1_2_1_1"><i><font color="Green" title="E57">A33</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E11:26_3_2_1_1_2_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_2_1_1_3"/><i><font color="Green" title="E59">A36</font></i>: 
<font color="Maroon" title="c19">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_2_1_1_3"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">G4</font> being   <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="E4:26_3_2_1_1_2_1_1_3"/><i><font color="Green" title="E60">A37</font></i>: 
( <font color="Maroon" title="c20">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E59">A36</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">G5</font> = <font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_1_2_1_1"><i><font color="Green" title="E53">A29</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_2_1_1_3"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_2_1_1_3"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_2_1_1_3"/><b>then </b><i><font color="Green" title="E61">A38</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_2_1_1_3"/><i><font color="Green" title="E62">A39</font></i>: 
<font color="Maroon" title="c21">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E63">A40</font></i>: <font color="Maroon" title="c17">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G5</font> = 
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E64">A41</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c19">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E60">A37</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_1_2_1_1"><i><font color="Green" title="E58">A35</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E64">A41</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_2_1_1_3"/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c19">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_1_2_1_1"><i><font color="Green" title="E55">A31</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E61">A38</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E62">A39</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_2_1_1_3"><i><font color="Green" title="E63">A40</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:26_3_2_1_1_2_1_1"/><b>then </b><i><font color="Green" title="E59">A42</font></i>: 
<font color="Maroon" title="c18">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E13:26_3_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">q'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_1_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">g1</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">q'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1"><i><font color="Green" title="E59">A42</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E15:26_3_2_1_1_2_1_1"/>
 ex <font color="Olive" title="b1">G1</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">q'</font> st <br/>( <font color="Olive" title="b1">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G1</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="c13">p1'</font> &amp; <font color="Olive" title="b1">G1</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="c14">q'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_2_1_1"><i><font color="Green" title="E54">A30</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_2_1_1"><i><font color="Green" title="E59">A42</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_1_2_1_2"/><i><font color="Green" title="E53">A43</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">g01</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_2_1_2"/><i><font color="Green" title="E54">A44</font></i>: 
( <font color="Maroon" title="c17">g01</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c17">g01</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="c13">p1'</font> &amp; <font color="Maroon" title="c17">g01</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="c13">p1'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4" title="BORSUK_2:th.4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_1_2_1_2"/>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c13">p1'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c13">p1'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_2"><i><font color="Green" title="E54">A44</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_1_2_1_2"/>
 ex <font color="Olive" title="b1">G1</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">q'</font> st <br/>( <font color="Olive" title="b1">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G1</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="c13">p1'</font> &amp; <font color="Olive" title="b1">G1</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="c14">q'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_2_1_2"><i><font color="Green" title="E53">A43</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_2_1_2"><i><font color="Green" title="E54">A44</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c17">G1</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c14">q'</font><b> such that </b><br/><a NAME="E18:26_3_2_1_1"/><i><font color="Green" title="E53">A45</font></i>: 
( <font color="Maroon" title="c17">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c17">G1</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="c13">p1'</font> &amp; <font color="Maroon" title="c17">G1</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="c14">q'</font> )
 ;<br/><a NAME="E19:26_3_2_1_1"/><i><font color="Green" title="E54">A46</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green" title="E37">A12</font></i></a>, <a class="ref" href="jordan20.html#T22" target="_self" title="JORDAN20:th.22">Th22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">f2</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_1"/><i><font color="Green" title="E55">A47</font></i>: 
( <font color="Maroon" title="c18">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c18">f2</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="c9">q</font> &amp; <font color="Maroon" title="c18">f2</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="c6">q2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E22:26_3_2_1_1"/><i><font color="Green" title="E56">A48</font></i>: 
<font color="Maroon" title="c18">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E23:26_3_2_1_1"/><i><font color="Green" title="E57">A49</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E58">A50</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E25:26_3_2_1_1"/><i><font color="Green" title="E59">A51</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E26:26_3_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="E1:26_3_2_1_1_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_4"/><i><font color="Green" title="E60">A52</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_4"/><i><font color="Green" title="E61">A53</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_4"><i><font color="Green" title="E60">A52</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_1_4_1"/><i><font color="Green" title="E62">A54</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font>
 ;<br/><a NAME="E2:26_3_2_1_1_4_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_1_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_4"><i><font color="Green" title="E60">A52</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_1_4_1"><i><font color="Green" title="E62">A54</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:26_3_2_1_1_4"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_4"><i><font color="Green" title="E61">A53</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="E7:26_3_2_1_1_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:26_3_2_1_1"/><b>then </b><i><font color="Green" title="E60">A55</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c18">f2</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E23:26_3_2_1_1"><i><font color="Green" title="E57">A49</font></i></a>, <a class="txt" href="jordan20.html#E24:26_3_2_1_1"><i><font color="Green" title="E58">A50</font></i></a>, <a class="txt" href="jordan20.html#E25:26_3_2_1_1"><i><font color="Green" title="E59">A51</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E29:26_3_2_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_5"/><i><font color="Green" title="E61">A56</font></i>: 
<font color="Maroon" title="c20">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_5"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">G4</font> being   <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="E4:26_3_2_1_1_5"/><i><font color="Green" title="E62">A57</font></i>: 
( <font color="Maroon" title="c21">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_5"><i><font color="Green" title="E61">A56</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">G5</font> = <font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_5"/>
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:26_3_2_1_1"><i><font color="Green" title="E54">A46</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_5"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_5"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_5"/><b>then </b><i><font color="Green" title="E63">A58</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_5"/><i><font color="Green" title="E64">A59</font></i>: 
<font color="Maroon" title="c22">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_5"><i><font color="Green" title="E62">A57</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E65">A60</font></i>: <font color="Maroon" title="c18">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E66">A61</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">g2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_5"><i><font color="Green" title="E62">A57</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">g2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E27:26_3_2_1_1"><i><font color="Green" title="E60">A55</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_5"><i><font color="Green" title="E66">A61</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_5"/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E22:26_3_2_1_1"><i><font color="Green" title="E56">A48</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_5"><i><font color="Green" title="E63">A58</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_5"><i><font color="Green" title="E64">A59</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_5"><i><font color="Green" title="E65">A60</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:26_3_2_1_1"/><b>then </b><i><font color="Green" title="E61">A62</font></i>: 
<font color="Maroon" title="c19">g2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E31:26_3_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">q2'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">G2</font> = <font color="Maroon" title="c19">g2</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c15">q2'</font> <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_1"><i><font color="Green" title="E61">A62</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><a NAME="E33:26_3_2_1_1"/><i><font color="Green" title="E62">A63</font></i>: 
( <font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font></span>)</span> <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="c13">p1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font></span>)</span> <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="c15">q2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E18:26_3_2_1_1"><i><font color="Green" title="E53">A45</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_1"><i><font color="Green" title="E55">A47</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_1"><i><font color="Green" title="E61">A62</font></i></a>, <a class="txt" href="jordan20.html#E33"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_1_6_1"/>
( <font color="Maroon" title="c6">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> or <font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_1_6_1_1"/>
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E63">A64</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="ref" href="jordan20.html#T22" target="_self" title="JORDAN20:th.22">Th22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">f3</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_1_6_1_1"/><i><font color="Green" title="E64">A65</font></i>: 
( <font color="Maroon" title="c21">f3</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c21">f3</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="c6">q2</font> &amp; <font color="Maroon" title="c21">f3</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">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_1_6_1_1"/><i><font color="Green" title="E65">A66</font></i>: 
<font color="Maroon" title="c21">f3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_1_6_1_1"/><i><font color="Green" title="E66">A67</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E67">A68</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">f3</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_1_6_1_1"/><i><font color="Green" title="E68">A69</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E9:26_3_2_1_1_6_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">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="E1:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_1_6_1_1_2"/><i><font color="Green" title="E69">A70</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c23">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c23">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_1_2"><i><font color="Green" title="E69">A70</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_1_6_1_1_2"/><b>then </b>
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_1_2"><i><font color="Green" title="E69">A70</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_1_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E69">A71</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">g3</font> = <font color="Maroon" title="c21">f3</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1_6_1_1"><i><font color="Green" title="E66">A67</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_1_6_1_1"><i><font color="Green" title="E67">A68</font></i></a>, <a class="txt" href="jordan20.html#E8:26_3_2_1_1_6_1_1"><i><font color="Green" title="E68">A69</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:26_3_2_1_1_6_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_6_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_1_6_1_1_3"/><i><font color="Green" title="E70">A72</font></i>: 
<font color="Maroon" title="c23">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_1_6_1_1_3"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_1"><i><font color="Green" title="E47">A22</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">G4</font> being   <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="E4:26_3_2_1_1_6_1_1_3"/><i><font color="Green" title="E71">A73</font></i>: 
( <font color="Maroon" title="c24">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c23">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E70">A72</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">G5</font> = <font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_1_6_1_1_3"/>
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_1_6_1_1"><i><font color="Green" title="E63">A64</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_1_6_1_1_3"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_1_6_1_1_3"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_1_6_1_1_3"/><b>then </b><i><font color="Green" title="E72">A74</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_1_6_1_1_3"/><i><font color="Green" title="E73">A75</font></i>: 
<font color="Maroon" title="c25">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E71">A73</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E74">A76</font></i>: <font color="Maroon" title="c21">f3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c25">G5</font> = 
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E75">A77</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">g3</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c23">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E71">A73</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">g3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_1_6_1_1"><i><font color="Green" title="E69">A71</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E75">A77</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_1_6_1_1_3"/>
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c23">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_1_6_1_1"><i><font color="Green" title="E65">A66</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E72">A74</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E73">A75</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_1_6_1_1_3"><i><font color="Green" title="E74">A76</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:26_3_2_1_1_6_1_1"/><b>then </b><i><font color="Green" title="E70">A78</font></i>: 
<font color="Maroon" title="c22">g3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E14:26_3_2_1_1_6_1_1"/><b>then </b>
<font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c16">p2'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><a NAME="E15:26_3_2_1_1_6_1_1"/><b>then </b>
<font color="Maroon" title="c22">g3</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c16">p2'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1_6_1_1"><i><font color="Green" title="E70">A78</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E16:26_3_2_1_1_6_1_1"/>
 ex <font color="Olive" title="b1">G3</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c16">p2'</font> st <br/>( <font color="Olive" title="b1">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G3</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="c15">q2'</font> &amp; <font color="Olive" title="b1">G3</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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_1_6_1_1"><i><font color="Green" title="E64">A65</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_1_6_1_1"><i><font color="Green" title="E70">A78</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_6_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_1_6_1_2"/><i><font color="Green" title="E63">A79</font></i>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c21">g01</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_1_6_1_2"/><i><font color="Green" title="E64">A80</font></i>: 
( <font color="Maroon" title="c21">g01</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c21">g01</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="c15">q2'</font> &amp; <font color="Maroon" title="c21">g01</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="c15">q2'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4" title="BORSUK_2:th.4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_1_6_1_2"/>
<font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c15">q2'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_6_1_2"/><b>then </b>
<font color="Maroon" title="c21">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c15">q2'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_2"><i><font color="Green" title="E64">A80</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_1_6_1_2"/>
 ex <font color="Olive" title="b1">G3</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c16">p2'</font> st <br/>( <font color="Olive" title="b1">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G3</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="c15">q2'</font> &amp; <font color="Olive" title="b1">G3</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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_1_6_1_2"><i><font color="Green" title="E63">A79</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_1_6_1_2"><i><font color="Green" title="E64">A80</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c21">G3</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q2'</font>,<font color="Maroon" title="c16">p2'</font><b> such that </b><br/><a NAME="E36:26_3_2_1_1"/><i><font color="Green" title="E63">A81</font></i>: 
( <font color="Maroon" title="c21">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c21">G3</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="c15">q2'</font> &amp; <font color="Maroon" title="c21">G3</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="c16">p2'</font> )
 ;<br/><a NAME="E37:26_3_2_1_1"/>
( <span class="p1">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c21">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c21">G3</font></span>)</span> <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="c13">p1'</font> &amp; <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">G1</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c20">G2</font></span>)</span> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c21">G3</font></span>)</span> <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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:26_3_2_1_1"><i><font color="Green" title="E62">A63</font></i></a>, <a class="txt" href="jordan20.html#E36:26_3_2_1_1"><i><font color="Green" title="E63">A81</font></i></a>, <a class="txt" href="jordan20.html#E33"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E38:26_3_2_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="jordan20.html#E7:26"><i><font color="Green" title="E34">A7</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_1"><i><font color="Green" title="E48">A23</font></i></a>, <a class="ref" href="jordan20.html#T18" target="_self" title="JORDAN20:th.18">Th18</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c4">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p3</font> where <font color="Olive" title="b1">p3</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Olive" title="b1">p3</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p3</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then consider </b><font color="Maroon" title="c10">p3</font> being   <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><b> such that </b><br/><a NAME="E4:26_3_2_1_2"/><i><font color="Green" title="E45">A82</font></i>: 
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">p3</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c10">p3</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/><a NAME="E5:26_3_2_1_2"/><i><font color="Green" title="E46">A83</font></i>: 
not <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E6:26_3_2_1_2"/><i><font color="Green" title="E47">A84</font></i>: 
<font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">P</font>
 <b>by </b><i><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c11">Qa</font> = <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> as  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"><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="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3"><i><font color="Green" title="E40">A15</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3_2_1_2"><i><font color="Green" title="E46">A83</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c12">Qa'</font> = <font color="Maroon" title="c11">Qa</font> as   <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/><div><i><font color="Green" title="E48">A85</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_2_1"/><i><font color="Green" title="E49">A86</font></i>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 ;<br/><a NAME="E2:26_3_2_1_2_1"/><b>then </b>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2"><i><font color="Green" title="E45">A82</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_2_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E6:26"><i><font color="Green" title="E33">A6</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_2_1"><i><font color="Green" title="E49">A86</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><a NAME="E10:26_3_2_1_2"/><i><font color="Green" title="E49">A87</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E11:26_3_2_1_2"/><i><font color="Green" title="E50">A88</font></i>: 
not <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E12:26_3_2_1_2"/><i><font color="Green" title="E51">A89</font></i>: 
not <font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26"><i><font color="Green" title="E35">A9</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E13:26_3_2_1_2"/><i><font color="Green" title="E52">A90</font></i>: 
not <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_2"/>
not <font color="Maroon" title="c9">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green" title="E37">A12</font></i></a>, <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="c13">p1'</font> = <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c14">q'</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c15">q1'</font> = <font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c16">p2'</font> = <font color="Maroon" title="c4">p2</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="jordan20.html#E4:26"><i><font color="Green" title="E31">A4</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2"><i><font color="Green" title="E49">A87</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2"><i><font color="Green" title="E50">A88</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2"><i><font color="Green" title="E51">A89</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2"><i><font color="Green" title="E52">A90</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_2_2_1"/>
( <font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> or <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_2_2_1_1"/>
<font color="Maroon" title="c9">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E53">A91</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="jordan20.html#T22" target="_self" title="JORDAN20:th.22">Th22</a></i>;<br/><b>then consider </b><font color="Maroon" title="c17">f1</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_2_2_1_1"/><i><font color="Green" title="E54">A92</font></i>: 
( <font color="Maroon" title="c17">f1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c17">f1</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="c9">q</font> &amp; <font color="Maroon" title="c17">f1</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">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_2_2_1_1"/><i><font color="Green" title="E55">A93</font></i>: 
<font color="Maroon" title="c17">f1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_2_2_1_1"/><i><font color="Green" title="E56">A94</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E57">A95</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_2_2_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c18">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="E1:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c19">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_2_1_1_2"/><i><font color="Green" title="E58">A96</font></i>: 
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c19">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c19">p</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2"><i><font color="Green" title="E45">A82</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_1_2"><i><font color="Green" title="E58">A96</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_2_2_1_1_2"/><b>then </b>
( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_1_2"><i><font color="Green" title="E58">A96</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_2_2_1_1_2"/>
<font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E9:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E58">A97</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c18">g1</font> = <font color="Maroon" title="c17">f1</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_2"><i><font color="Green" title="E49">A87</font></i></a>, <a class="txt" href="jordan20.html#E6:26_3_2_1_2_2_1_1"><i><font color="Green" title="E56">A94</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_2_2_1_1"><i><font color="Green" title="E57">A95</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E11:26_3_2_1_2_2_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_2_1_1_3"/><i><font color="Green" title="E59">A98</font></i>: 
<font color="Maroon" title="c19">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_2_1_1_3"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green" title="E47">A84</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">G4</font> being   <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="E4:26_3_2_1_2_2_1_1_3"/><i><font color="Green" title="E60">A99</font></i>: 
( <font color="Maroon" title="c20">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c19">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E59">A98</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c21">G5</font> = <font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_2_1_1_3"/>
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_2_2_1_1"><i><font color="Green" title="E53">A91</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_2_1_1_3"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_2_1_1_3"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_2_1_1_3"/><b>then </b><i><font color="Green" title="E61">A100</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_2_1_1_3"/><i><font color="Green" title="E62">A101</font></i>: 
<font color="Maroon" title="c21">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E60">A99</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E63">A102</font></i>: <font color="Maroon" title="c17">f1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G5</font> = 
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E64">A103</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">g1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c19">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E60">A99</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">g1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c4">p2</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E9:26_3_2_1_2_2_1_1"><i><font color="Green" title="E58">A97</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E64">A103</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_2_1_1_3"/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c19">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_2_2_1_1"><i><font color="Green" title="E55">A93</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E61">A100</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E62">A101</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_2_1_1_3"><i><font color="Green" title="E63">A102</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E12:26_3_2_1_2_2_1_1"/><b>then </b><i><font color="Green" title="E59">A104</font></i>: 
<font color="Maroon" title="c18">g1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E13:26_3_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c16">p2'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><a NAME="E14:26_3_2_1_2_2_1_1"/><b>then </b>
<font color="Maroon" title="c18">g1</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c16">p2'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1"><i><font color="Green" title="E59">A104</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E15:26_3_2_1_2_2_1_1"/>
 ex <font color="Olive" title="b1">G1</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c16">p2'</font> st <br/>( <font color="Olive" title="b1">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G1</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="c14">q'</font> &amp; <font color="Olive" title="b1">G1</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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_2_1_1"><i><font color="Green" title="E54">A92</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_2_1_1"><i><font color="Green" title="E59">A104</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_2_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_2_1_2"/><i><font color="Green" title="E53">A105</font></i>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c17">g01</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_2_1_2"/><i><font color="Green" title="E54">A106</font></i>: 
( <font color="Maroon" title="c17">g01</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c17">g01</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="c16">p2'</font> &amp; <font color="Maroon" title="c17">g01</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="c16">p2'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4" title="BORSUK_2:th.4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_2_2_1_2"/>
<font color="Maroon" title="c16">p2'</font>,<font color="Maroon" title="c16">p2'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_2_1_2"/><b>then </b>
<font color="Maroon" title="c17">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c16">p2'</font>,<font color="Maroon" title="c16">p2'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_2"><i><font color="Green" title="E54">A106</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_2_2_1_2"/>
 ex <font color="Olive" title="b1">G1</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c16">p2'</font> st <br/>( <font color="Olive" title="b1">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G1</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="c14">q'</font> &amp; <font color="Olive" title="b1">G1</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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_2_1_2"><i><font color="Green" title="E53">A105</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_2_1_2"><i><font color="Green" title="E54">A106</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c17">G1</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c14">q'</font>,<font color="Maroon" title="c16">p2'</font><b> such that </b><br/><a NAME="E18:26_3_2_1_2"/><i><font color="Green" title="E53">A107</font></i>: 
( <font color="Maroon" title="c17">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c17">G1</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="c14">q'</font> &amp; <font color="Maroon" title="c17">G1</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="c16">p2'</font> )
 ;<br/><a NAME="E19:26_3_2_1_2"/><i><font color="Green" title="E54">A108</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3"><i><font color="Green" title="E38">A13</font></i></a>, <a class="txt" href="jordan20.html#E26"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c18">f2</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E21:26_3_2_1_2"/><i><font color="Green" title="E55">A109</font></i>: 
( <font color="Maroon" title="c18">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c18">f2</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="c5">q1</font> &amp; <font color="Maroon" title="c18">f2</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="c9">q</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E22:26_3_2_1_2"/><i><font color="Green" title="E56">A110</font></i>: 
<font color="Maroon" title="c18">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E23:26_3_2_1_2"/><i><font color="Green" title="E57">A111</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c18">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E58">A112</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c18">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E25:26_3_2_1_2"/><i><font color="Green" title="E59">A113</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E26:26_3_2_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c19">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="E1:26_3_2_1_2_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c20">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_4"/><i><font color="Green" title="E60">A114</font></i>: 
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c20">p</font>,<font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_4"/><i><font color="Green" title="E61">A115</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_4"><i><font color="Green" title="E60">A114</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:26_3_2_1_2_4_1"/><i><font color="Green" title="E62">A116</font></i>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 ;<br/><a NAME="E2:26_3_2_1_2_4_1"/>
 <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c9">q</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></i>;<br/><b>hence </b><a NAME="E3:26_3_2_1_2_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3"><i><font color="Green" title="E36">A11</font></i></a>, <a class="txt" href="jordan20.html#E5:26_3"><i><font color="Green" title="E37">A12</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_4"><i><font color="Green" title="E60">A114</font></i></a>, <a class="txt" href="jordan20.html#E1:26_3_2_1_2_4_1"><i><font color="Green" title="E62">A116</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:26_3_2_1_2_4"/><b>then </b>
( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_4"><i><font color="Green" title="E61">A115</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="E7:26_3_2_1_2_4"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E27:26_3_2_1_2"/><b>then </b><i><font color="Green" title="E60">A117</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c18">f2</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E23:26_3_2_1_2"><i><font color="Green" title="E57">A111</font></i></a>, <a class="txt" href="jordan20.html#E24:26_3_2_1_2"><i><font color="Green" title="E58">A112</font></i></a>, <a class="txt" href="jordan20.html#E25:26_3_2_1_2"><i><font color="Green" title="E59">A113</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E29:26_3_2_1_2"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c20">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_5"/><i><font color="Green" title="E61">A118</font></i>: 
<font color="Maroon" title="c20">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_5"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green" title="E47">A84</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c21">G4</font> being   <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="E4:26_3_2_1_2_5"/><i><font color="Green" title="E62">A119</font></i>: 
( <font color="Maroon" title="c21">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c20">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_5"><i><font color="Green" title="E61">A118</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c22">G5</font> = <font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_5"/>
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:26_3_2_1_2"><i><font color="Green" title="E54">A108</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_5"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_5"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_5"/><b>then </b><i><font color="Green" title="E63">A120</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_5"/><i><font color="Green" title="E64">A121</font></i>: 
<font color="Maroon" title="c22">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_5"><i><font color="Green" title="E62">A119</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E65">A122</font></i>: <font color="Maroon" title="c18">f2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c21">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E66">A123</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">g2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_5"><i><font color="Green" title="E62">A119</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c19">g2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c21">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c2">Q1</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c9">q</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E27:26_3_2_1_2"><i><font color="Green" title="E60">A117</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_5"><i><font color="Green" title="E66">A123</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_5"/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c20">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E22:26_3_2_1_2"><i><font color="Green" title="E56">A110</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_5"><i><font color="Green" title="E63">A120</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_5"><i><font color="Green" title="E64">A121</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_5"><i><font color="Green" title="E65">A122</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E30:26_3_2_1_2"/><b>then </b><i><font color="Green" title="E61">A124</font></i>: 
<font color="Maroon" title="c19">g2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E31:26_3_2_1_2"/><b>then </b>
<font color="Maroon" title="c15">q1'</font>,<font color="Maroon" title="c14">q'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c20">G2</font> = <font color="Maroon" title="c19">g2</font> as    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q1'</font>,<font color="Maroon" title="c14">q'</font> <b>by </b><i><a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_2"><i><font color="Green" title="E61">A124</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><a NAME="E33:26_3_2_1_2"/><i><font color="Green" title="E62">A125</font></i>: 
( <font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font></span>)</span> <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="c15">q1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font></span>)</span> <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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E18:26_3_2_1_2"><i><font color="Green" title="E53">A107</font></i></a>, <a class="txt" href="jordan20.html#E21:26_3_2_1_2"><i><font color="Green" title="E55">A109</font></i></a>, <a class="txt" href="jordan20.html#E30:26_3_2_1_2"><i><font color="Green" title="E61">A124</font></i></a>, <a class="txt" href="jordan20.html#E33"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_6_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:26_3_2_1_2_6_1"/>
( <font color="Maroon" title="c5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> or <font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_6_1_1"><b>case </b></a><a NAME="E1:26_3_2_1_2_6_1_1"/>
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font>
 ;<br/><div class="add"><a NAME="E2:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green" title="E63">A126</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E2:26"><i><font color="Green" title="E29">A2</font></i></a>, <a class="txt" href="jordan20.html#E26"><i><font color="Green" title="E21">Lm1</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c21">f3</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span><b> such that </b><br/><a NAME="E4:26_3_2_1_2_6_1_1"/><i><font color="Green" title="E64">A127</font></i>: 
( <font color="Maroon" title="c21">f3</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c21">f3</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">p1</font> &amp; <font color="Maroon" title="c21">f3</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="c5">q1</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/><a NAME="E5:26_3_2_1_2_6_1_1"/><i><font color="Green" title="E65">A128</font></i>: 
<font color="Maroon" title="c21">f3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E6:26_3_2_1_2_6_1_1"/><i><font color="Green" title="E66">A129</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c21">f3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> 
 <b>by </b><i><a class="txt" href="jordan20.html#E8:26_3"><i><font color="Green" title="E39">A14</font></i></a>, <a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E67">A130</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c21">f3</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E8:26_3_2_1_2_6_1_1"/><i><font color="Green" title="E68">A131</font></i>: 
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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">Qa</font>
 <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="E9:26_3_2_1_2_6_1_1"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c22">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="E1:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p</font> where <font color="Olive" title="b1">p</font> is   <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> : (  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Olive" title="b1">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b1">p</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c23">p</font> being   <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><b> such that </b><br/><a NAME="E3:26_3_2_1_2_6_1_1_2"/><i><font color="Green" title="E69">A132</font></i>: 
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c23">p</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c23">p</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Maroon" title="c23">p</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>
<a NAME="E4:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q2</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E5:26"><i><font color="Green" title="E32">A5</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_1_2"><i><font color="Green" title="E69">A132</font></i></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></i>;<br/>
<a NAME="E5:26_3_2_1_2_6_1_1_2"/><b>then </b>
( <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c6">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_1_2"><i><font color="Green" title="E69">A132</font></i></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E6:26_3_2_1_2_6_1_1_2"/>
<font color="Maroon" title="c22">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>


</div><b>end;</b></div><a NAME="E10:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green" title="E69">A133</font></i>: 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c11">Qa</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T29" title="JORDAN6:th.29">JORDAN6:29</a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c22">g3</font> = <font color="Maroon" title="c21">f3</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"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2_6_1_1"><i><font color="Green" title="E66">A129</font></i></a>, <a class="txt" href="jordan20.html#E7:26_3_2_1_2_6_1_1"><i><font color="Green" title="E67">A130</font></i></a>, <a class="txt" href="jordan20.html#E8:26_3_2_1_2_6_1_1"><i><font color="Green" title="E68">A131</font></i></a>, <a class="ref" href="funct_2.html#T4" title="FUNCT_2:th.4">FUNCT_2:4</a></i>;<br/><a NAME="E12:26_3_2_1_2_6_1_1"/>
for <font color="Olive" title="b1">G</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>  st <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> holds <br/><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_6_1_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c23">G</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:26_3_2_1_2_6_1_1_3"/><i><font color="Green" title="E70">A134</font></i>: 
<font color="Maroon" title="c23">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 ;<br/>

<a NAME="E2:26_3_2_1_2_6_1_1_3"/>
<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<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">Qa'</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E6:26_3_2_1_2"><i><font color="Green" title="E47">A84</font></i></a>, <a class="ref" href="pre_topc.html#T28" title="PRE_TOPC:th.28">PRE_TOPC:28</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c24">G4</font> being   <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="E4:26_3_2_1_2_6_1_1_3"/><i><font color="Green" title="E71">A135</font></i>: 
( <font color="Maroon" title="c24">G4</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> &amp; <font color="Maroon" title="c23">G</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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">Qa'</font></span>)</span></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E70">A134</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c25">G5</font> = <font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p2">(<span class="default"><span class="p3">(<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> <span class="p3">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span></span>)</span> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> ;<br/>
<a NAME="E6:26_3_2_1_2_6_1_1_3"/>
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:26_3_2_1_2_6_1_1"><i><font color="Green" title="E63">A126</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<a NAME="E7:26_3_2_1_2_6_1_1_3"/><b>then </b>
not  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E8:26_3_2_1_2_6_1_1_3"/><b>then </b>
not 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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span> is <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>
 
;<br/>
<a NAME="E9:26_3_2_1_2_6_1_1_3"/><b>then </b><i><font color="Green" title="E72">A136</font></i>: 
not <span class="p1">(<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> <span class="p1">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span> is <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a>
 
<b>by </b><i><a class="ref" href="struct_0.html#D1" title="STRUCT_0:def.1">STRUCT_0:def 1</a></i>;<br/>
<a NAME="E10:26_3_2_1_2_6_1_1_3"/><i><font color="Green" title="E73">A137</font></i>: 
<font color="Maroon" title="c25">G5</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E71">A135</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<i><font color="Green" title="E74">A138</font></i>: <font color="Maroon" title="c21">f3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c25">G5</font> = 
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p1">(<span class="default"><font color="Maroon" title="c24">G4</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<i><font color="Green" title="E75">A139</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">g3</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
 <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c23">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p3">(<span class="default"><span class="p4">(<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">Qa'</font></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E71">A135</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c12">Qa'</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c22">g3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c12">Qa'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">G4</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c5">q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="jordan20.html#E10:26_3_2_1_2_6_1_1"><i><font color="Green" title="E69">A133</font></i></a>, <a class="txt" href="jordan20.html#E12:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E75">A139</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
;<br/>
<b>hence </b><a NAME="E14:26_3_2_1_2_6_1_1_3"/>
<font color="Maroon" title="c22">g3</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c23">G</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E5:26_3_2_1_2_6_1_1"><i><font color="Green" title="E65">A128</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E72">A136</font></i></a>, <a class="txt" href="jordan20.html#E10:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E73">A137</font></i></a>, <a class="txt" href="jordan20.html#E11:26_3_2_1_2_6_1_1_3"><i><font color="Green" title="E74">A138</font></i></a>, <a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/>


</div><b>end;</b></div><a NAME="E13:26_3_2_1_2_6_1_1"/><b>then </b><i><font color="Green" title="E70">A140</font></i>: 
<font color="Maroon" title="c22">g3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T55" title="TOPS_2:th.55">TOPS_2:55</a></i>;<br/><a NAME="E14:26_3_2_1_2_6_1_1"/><b>then </b>
<font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c15">q1'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></i>;<br/><a NAME="E15:26_3_2_1_2_6_1_1"/><b>then </b>
<font color="Maroon" title="c22">g3</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c15">q1'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2_6_1_1"><i><font color="Green" title="E70">A140</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E16:26_3_2_1_2_6_1_1"/>
 ex <font color="Olive" title="b1">G3</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c15">q1'</font> st <br/>( <font color="Olive" title="b1">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G3</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="c13">p1'</font> &amp; <font color="Olive" title="b1">G3</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="c15">q1'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:26_3_2_1_2_6_1_1"><i><font color="Green" title="E64">A127</font></i></a>, <a class="txt" href="jordan20.html#E13:26_3_2_1_2_6_1_1"><i><font color="Green" title="E70">A140</font></i></a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_6_1_2"><b>case </b></a><a NAME="E1:26_3_2_1_2_6_1_2"/><i><font color="Green" title="E63">A141</font></i>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ;<br/><div class="add"><b>consider </b><font color="Maroon" title="c21">g01</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"><span class="p3">(<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="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Qa</font></span>)</span><b> such that </b><br/><a NAME="E3:26_3_2_1_2_6_1_2"/><i><font color="Green" title="E64">A142</font></i>: 
( <font color="Maroon" title="c21">g01</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c21">g01</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="c15">q1'</font> &amp; <font color="Maroon" title="c21">g01</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="c15">q1'</font> )
 <b>by </b><i><a class="ref" href="borsuk_2.html#T4" title="BORSUK_2:th.4">BORSUK_2:4</a></i>;<br/><a NAME="E4:26_3_2_1_2_6_1_2"/>
<font color="Maroon" title="c15">q1'</font>,<font color="Maroon" title="c15">q1'</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_6_1_2"/><b>then </b>
<font color="Maroon" title="c21">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q1'</font>,<font color="Maroon" title="c15">q1'</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_2"><i><font color="Green" title="E64">A142</font></i></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></i>;<br/><b>hence </b><a NAME="E6:26_3_2_1_2_6_1_2"/>
 ex <font color="Olive" title="b1">G3</font> being   <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c15">q1'</font> st <br/>( <font color="Olive" title="b1">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Olive" title="b1">G3</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="c13">p1'</font> &amp; <font color="Olive" title="b1">G3</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="c15">q1'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26_3_2_1_2_6_1_2"><i><font color="Green" title="E63">A141</font></i></a>, <a class="txt" href="jordan20.html#E3:26_3_2_1_2_6_1_2"><i><font color="Green" title="E64">A142</font></i></a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>then consider </b><font color="Maroon" title="c21">G3</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p1'</font>,<font color="Maroon" title="c15">q1'</font><b> such that </b><br/><a NAME="E36:26_3_2_1_2"/><i><font color="Green" title="E63">A143</font></i>: 
( <font color="Maroon" title="c21">G3</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <font color="Maroon" title="c21">G3</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="c13">p1'</font> &amp; <font color="Maroon" title="c21">G3</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="c15">q1'</font> )
 ;<br/><a NAME="E37:26_3_2_1_2"/>
( <font color="Maroon" title="c21">G3</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font></span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">G3</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font></span>)</span></span>)</span> <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="c13">p1'</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c21">G3</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c20">G2</font> <a href="borsuk_2.html#K1" title="BORSUK_2:func.1">+</a> <font color="Maroon" title="c17">G1</font></span>)</span></span>)</span> <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="c16">p2'</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E33:26_3_2_1_2"><i><font color="Green" title="E62">A125</font></i></a>, <a class="txt" href="jordan20.html#E36:26_3_2_1_2"><i><font color="Green" title="E63">A143</font></i></a>, <a class="txt" href="jordan20.html#E33"><i><font color="Green" title="E26">Lm2</font></i></a></i>;<br/><b>hence </b><a NAME="E38:26_3_2_1_2"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:26"><i><font color="Green" title="E30">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:26"><i><font color="Green" title="E35">A9</font></i></a>, <a class="txt" href="jordan20.html#E9:26_3_2_1_2"><i><font color="Green" title="E48">A85</font></i></a>, <a class="ref" href="jordan20.html#T18" target="_self" title="JORDAN20:th.18">Th18</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E17:26_3"/>
contradiction
 ;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E11:26"/>
<font color="Maroon" title="c2">Q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="jordan6.html#K5" title="JORDAN6:func.5">Segment</a> <font color="Maroon" title="c1">P</font>,<font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font>,<font color="Maroon" title="c5">q1</font>,<font color="Maroon" title="c6">q2</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:26"><i><font color="Green" title="E28">A1</font></i></a>, <a class="txt" href="jordan20.html#E6:26"><i><font color="Green" title="E33">A6</font></i></a>, <a class="ref" href="jordan20.html#T21" target="_self" title="JORDAN20:th.21">Th21</a></i>;<br/>


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