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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</font>, <font color="Olive" title="b3">q1</font>, <font color="Olive" title="b4">q2</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>  st <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">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="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font> &amp;  <a href="jordan5c.html#R1" title="JORDAN5C:pred.1">LE</a> <font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>,<font color="Maroon" title="c1">P</font>,<font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">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> holds <br/><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="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font>,<font color="Olive" title="b3">q1</font>,<font color="Olive" title="b4">q2</font>)</span><br/><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <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> implies <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>) )</span><br/>



<span class="kw">assume </span><span class="kw">that </span><br/><a NAME="E1:26"/><span class="lab"><font color="Green" title="E27">A1</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E2:26"/><span class="lab"><font color="Green" title="E28">A2</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E3:26"/><span class="lab"><font color="Green" title="E29">A3</font></span>: 
 <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>
 <span class="kw">and </span><br/><a NAME="E4:26"/><span class="lab"><font color="Green" title="E30">A4</font></span>: 
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <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>)</span><br/>

<span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan16.html#T18" title="JORDAN16:th.18">JORDAN16:18</a></span>;<br/>
<a NAME="E6:26"/><span class="lab"><font color="Green" title="E31">A5</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>;<br/>
<a NAME="E7:26"/><span class="kw">then </span><span class="lab"><font color="Green" title="E32">A6</font></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="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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>;<br/>
<a NAME="E8:26"/><span class="lab"><font color="Green" title="E33">A7</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E34">A8</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_1"/><span class="lab"><font color="Green" title="E35">A9</font></span>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:26_1"><span class="lab"><font color="Green" title="E35">A9</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a>, <a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E10:26"/><span class="lab"><font color="Green" title="E35">A10</font></span>: 
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<div><span class="lab"><font color="Green" title="E36">A11</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_2"/><span class="lab"><font color="Green" title="E37">A12</font></span>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:26_2"><span class="lab"><font color="Green" title="E37">A12</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a>, <a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E12:26"/><span class="lab"><font color="Green" title="E37">A13</font></span>: 
( <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> )
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3"><span class="kw">now </span></a><div class="add"><a NAME="E1:26_3"/><span class="lab"><font color="Green" title="E38">A14</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E2:26_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E39">A15</font></span>: 
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan20.html#T22" target="_self" title="JORDAN20:th.22">Th23</a></span>;<br/><a NAME="E3:26_3"/><span class="lab"><font color="Green" title="E40">A16</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>;<br/><a NAME="E4:26_3"/><span class="lab"><font color="Green" title="E41">A17</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><a NAME="E5:26_3"/><span class="lab"><font color="Green" title="E42">A18</font></span>: 
 <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="E6:26_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="jordan16.html#T21" title="JORDAN16:th.21">JORDAN16:21</a></span>;<br/><a NAME="E7:26_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E43">A19</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/><span class="kw">assume </span><a NAME="E8:26_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">x8</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <span class="kw"> such that </span><br/><a NAME="E10:26_3"/><span class="lab"><font color="Green" title="E44">A20</font></span>: 
<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>
 <span class="kw">and </span><br/><a NAME="E11:26_3"/><span class="lab"><font color="Green" title="E45">A21</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></span>;<br/><span class="kw">reconsider </span><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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a></span>;<br/><a NAME="E13:26_3"/><span class="lab"><font color="Green" title="E46">A22</font></span>: 
<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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="ref" href="jordan16.html#T5" title="JORDAN16:th.5">JORDAN16:5</a></span>;<br/><a NAME="E14:26_3"/>
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="txt" href="#E1:26_3"><span class="lab"><font color="Green" title="E38">A14</font></span></a>, <a class="ref" href="jordan5c.html#T13" title="JORDAN5C:th.13">JORDAN5C:13</a></span>;<br/><span class="kw">then </span><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>)
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:26_3"><span class="lab"><font color="Green" title="E41">A17</font></span></a>, <a class="ref" href="jordan20.html#T22" target="_self" title="JORDAN20:th.22">Th23</a></span>
<br/>.= 
<font color="Maroon" title="c1">P</font>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="ref" href="jordan20.html#T24" target="_self" title="JORDAN20:th.24">Th25</a></span>
;<br/><a NAME="E16:26_3"/><span class="kw">then </span><span class="lab"><font color="Green" title="E47">A23</font></span>: 
( <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>) )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></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><span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E2:26_3"><span class="lab"><font color="Green" title="E39">A15</font></span></a>, <a class="txt" href="#E16:26_3"><span class="lab"><font color="Green" title="E47">A23</font></span></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_1"/><span class="lab"><font color="Green" title="E48">A24</font></span>: 
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:26_3_2_1_1"/><span class="lab"><font color="Green" title="E49">A25</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:26_3_2_1_1"/>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E3:26_3"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E5:26_3_2_1_1"/><span class="lab"><font color="Green" title="E50">A26</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">Qa9</font> = <font color="Maroon" title="c10">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/><a NAME="E7:26_3_2_1_1"/><span class="lab"><font color="Green" title="E51">A27</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><a NAME="E8:26_3_2_1_1"/><span class="lab"><font color="Green" title="E52">A28</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E7:26_3"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="jordan20.html#T21" target="_self" title="JORDAN20:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E10:26_3_2_1_1"/><span class="lab"><font color="Green" title="E53">A29</font></span>: 
<font color="Maroon" title="c12">f2</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E11:26_3_2_1_1"/><span class="lab"><font color="Green" title="E54">A30</font></span>: 
( <font color="Maroon" title="c12">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">q</font> &amp; <font color="Maroon" title="c12">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E55">A31</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E13:26_3_2_1_1"/><span class="lab"><font color="Green" title="E56">A32</font></span>: 
(  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> &amp;  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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E14:26_3_2_1_1"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1"><span class="lab"><font color="Green" title="E48">A24</font></span></a>, <a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E15:26_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A33</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p3</font> &amp;  <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> )
 ;<br/><div><span class="lab"><font color="Green" title="E58">A34</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_3_2_1_1_2"/><span class="lab"><font color="Green" title="E59">A35</font></span>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_3_2_1_1_2"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E15:26_3_2_1_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_3_2_1_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:26_3_2_1_1_2"><span class="lab"><font color="Green" title="E59">A35</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:26_3_2_1_1"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">p19</font> = <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c15">q29</font> = <font color="Maroon" title="c6">q2</font>, <font color="Maroon" title="c16">p29</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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E5:26_3_2_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E13:26_3_2_1_1"><span class="lab"><font color="Green" title="E56">A32</font></span></a>, <a class="txt" href="#E2:26_3_2_1_1"><span class="lab"><font color="Green" title="E49">A25</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_3_2_1_1_3_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_3_1_1"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_1_3_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">p19</font>,<font color="Maroon" title="c14">q9</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q9</font> )</span><br/><div class="add"><a NAME="E2:26_3_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A36</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="jordan16.html#T24" title="JORDAN16:th.24">JORDAN16:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E60">A37</font></span>: 
<font color="Maroon" title="c17">f1</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E61">A38</font></span>: 
( <font color="Maroon" title="c17">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E62">A39</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E7:26_3_2_1_1_3_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_3_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_1_3_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A40</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<a NAME="E3:26_3_2_1_1_3_1_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E13:26_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E15:26_3_2_1_1"><span class="lab"><font color="Green" title="E57">A33</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>;<br/>
<a NAME="E4:26_3_2_1_1_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A41</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_1_3_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="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E63">A40</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_1_3_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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_3_1_1_2"><span class="lab"><font color="Green" title="E64">A41</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:26_3_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A42</font></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="c9">q</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E9:26_3_2_1_1_3_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E4:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1"><span class="lab"><font color="Green" title="E50">A26</font></span></a>, <a class="txt" href="#E6:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E62">A39</font></span></a>, <a class="txt" href="#E8:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E11:26_3_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E64">A43</font></span>: 
<font color="Maroon" title="c17">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E12:26_3_2_1_1_3_1_1"/><span class="lab"><font color="Green" title="E65">A44</font></span>: 
 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="c10">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#K8" title="RELSET_1:func.8">"</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_3_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E66">A45</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_1_3_1_1_3"/>
<font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A46</font></span>: 
<font color="Maroon" title="c20">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E68">A47</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E66">A45</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="E7:26_3_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E69">A48</font></span>: 
<font color="Maroon" title="c21">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E67">A46</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A49</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E60">A37</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_1_3_1_1_3"/><span class="lab"><font color="Green" title="E71">A50</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E59">A36</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">A51</font></span>: <font color="Maroon" title="c17">f1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c21">G5</font> = 
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E68">A47</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E63">A42</font></span></a>, <a class="txt" href="#E8:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E70">A49</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_1_3_1_1_3"/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E64">A43</font></span></a>, <a class="txt" href="#E9:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E71">A50</font></span></a>, <a class="txt" href="#E7:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E69">A48</font></span></a>, <a class="txt" href="#E10:26_3_2_1_1_3_1_1_3"><span class="lab"><font color="Green" title="E72">A51</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:26_3_2_1_1_3_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E14:26_3_2_1_1_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A52</font></span>: 
<font color="Maroon" title="c18">g1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E65">A44</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E15:26_3_2_1_1_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">p19</font>,<font color="Maroon" title="c14">q9</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><a NAME="E16:26_3_2_1_1_3_1_1"/><span class="kw">then </span>
<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">p19</font>,<font color="Maroon" title="c14">q9</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="txt" href="#E14:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A52</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E17:26_3_2_1_1_3_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">p19</font>,<font color="Maroon" title="c14">q9</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E61">A38</font></span></a>, <a class="txt" href="#E14:26_3_2_1_1_3_1_1"><span class="lab"><font color="Green" title="E66">A52</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_3_1_2"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E59">A53</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">p19</font>,<font color="Maroon" title="c14">q9</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q9</font> )</span><br/><div class="add"><span class="kw">consider </span><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="c10">Qa</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_3_2_1_1_3_1_2"/><span class="lab"><font color="Green" title="E60">A54</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">p19</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#T3" title="BORSUK_2:th.3">BORSUK_2:3</a></span>;<br/><a NAME="E4:26_3_2_1_1_3_1_2"/>
<font color="Maroon" title="c13">p19</font>,<font color="Maroon" title="c13">p19</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_3_1_2"/><span class="kw">then </span>
<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">p19</font>,<font color="Maroon" title="c13">p19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_3_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E60">A54</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_3_2_1_1_3_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">p19</font>,<font color="Maroon" title="c14">q9</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q9</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E59">A53</font></span></a>, <a class="txt" href="#E3:26_3_2_1_1_3_1_2"><span class="lab"><font color="Green" title="E60">A54</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><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">p19</font>,<font color="Maroon" title="c14">q9</font><span class="kw"> such that </span><br/><a NAME="E21:26_3_2_1_1"/><span class="lab"><font color="Green" title="E59">A55</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q9</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_3_2_1_1_4_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_4_1_1"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_1_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">q29</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</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">p29</font> )</span><br/><div class="add"><a NAME="E2:26_3_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A56</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="ref" href="jordan20.html#T21" target="_self" title="JORDAN20:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E61">A57</font></span>: 
<font color="Maroon" title="c18">f3</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E62">A58</font></span>: 
( <font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font> &amp; <font color="Maroon" title="c18">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E63">A59</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E7:26_3_2_1_1_4_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_4_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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="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>  or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_1_4_1_1_2"/>
<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="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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A60</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<a NAME="E3:26_3_2_1_1_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">q1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a>, <a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>;<br/>
<a NAME="E4:26_3_2_1_1_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A61</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_1_4_1_1_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E64">A60</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_1_4_1_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_4_1_1_2"><span class="lab"><font color="Green" title="E65">A61</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:26_3_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A62</font></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="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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E9:26_3_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E65">A63</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><a NAME="E10:26_3_2_1_1_4_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E4:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">g3</font> = <font color="Maroon" title="c18">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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E63">A59</font></span></a>, <a class="txt" href="#E9:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E65">A63</font></span></a>, <a class="txt" href="#E8:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E12:26_3_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E66">A64</font></span>: 
<font color="Maroon" title="c18">f3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E13:26_3_2_1_1_4_1_1"/><span class="lab"><font color="Green" title="E67">A65</font></span>: 
 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="c10">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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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_4_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_1_4_1_1_3"/><span class="lab"><font color="Green" title="E68">A66</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_1_4_1_1_3"/>
<font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_1_4_1_1_3"/><span class="lab"><font color="Green" title="E69">A67</font></span>: 
<font color="Maroon" title="c21">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_1_4_1_1_3"/><span class="lab"><font color="Green" title="E70">A68</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E68">A66</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="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="E7:26_3_2_1_1_4_1_1_3"/><span class="lab"><font color="Green" title="E71">A69</font></span>: 
<font color="Maroon" title="c22">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E69">A67</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">A70</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E61">A57</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_1_4_1_1_3"/><span class="lab"><font color="Green" title="E73">A71</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E60">A56</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E74">A72</font></span>: <font color="Maroon" title="c18">f3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="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 class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E70">A68</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">g3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E64">A62</font></span></a>, <a class="txt" href="#E8:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E72">A70</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_1_4_1_1_3"/>
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E66">A64</font></span></a>, <a class="txt" href="#E9:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E73">A71</font></span></a>, <a class="txt" href="#E7:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E71">A69</font></span></a>, <a class="txt" href="#E10:26_3_2_1_1_4_1_1_3"><span class="lab"><font color="Green" title="E74">A72</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:26_3_2_1_1_4_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E15:26_3_2_1_1_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A73</font></span>: 
<font color="Maroon" title="c19">g3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E67">A65</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E16:26_3_2_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c15">q29</font>,<font color="Maroon" title="c16">p29</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><a NAME="E17:26_3_2_1_1_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">g3</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q29</font>,<font color="Maroon" title="c16">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="txt" href="#E15:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A73</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E18:26_3_2_1_1_4_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">q29</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</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">p29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E62">A58</font></span></a>, <a class="txt" href="#E15:26_3_2_1_1_4_1_1"><span class="lab"><font color="Green" title="E68">A73</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_1_4_1_2"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_1_4_1_2"/><span class="lab"><font color="Green" title="E60">A74</font></span>: 
<font color="Maroon" title="c6">q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">q29</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</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">p29</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">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="c10">Qa</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_3_2_1_1_4_1_2"/><span class="lab"><font color="Green" title="E61">A75</font></span>: 
( <font color="Maroon" title="c18">g01</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c18">g01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</font> &amp; <font color="Maroon" title="c18">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">q29</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#T3" title="BORSUK_2:th.3">BORSUK_2:3</a></span>;<br/><a NAME="E4:26_3_2_1_1_4_1_2"/>
<font color="Maroon" title="c15">q29</font>,<font color="Maroon" title="c15">q29</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_1_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q29</font>,<font color="Maroon" title="c15">q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_3_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E61">A75</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_3_2_1_1_4_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">q29</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</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">p29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E60">A74</font></span></a>, <a class="txt" href="#E3:26_3_2_1_1_4_1_2"><span class="lab"><font color="Green" title="E61">A75</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">G3</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q29</font>,<font color="Maroon" title="c16">p29</font><span class="kw"> such that </span><br/><a NAME="E24:26_3_2_1_1"/><span class="lab"><font color="Green" title="E60">A76</font></span>: 
( <font color="Maroon" title="c18">G3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c18">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q29</font> &amp; <font color="Maroon" title="c18">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">p29</font> )
 ;<br/><a NAME="E25:26_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_1_5"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A77</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_1_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_3_2_1_1_5_1"/><span class="lab"><font color="Green" title="E62">A78</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_3_2_1_1_5_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_3_2_1_1_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E13:26_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="txt" href="#E2:26_3_2_1_1_5"><span class="lab"><font color="Green" title="E61">A77</font></span></a>, <a class="txt" href="#E1:26_3_2_1_1_5_1"><span class="lab"><font color="Green" title="E62">A78</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:26_3_2_1_1_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A79</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_1_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_1_5"><span class="lab"><font color="Green" title="E61">A77</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_1_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E4:26_3_2_1_1_5"><span class="lab"><font color="Green" title="E62">A79</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:26_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A80</font></span>: 
 <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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E27:26_3_2_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E10:26_3_2_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c12">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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_1"><span class="lab"><font color="Green" title="E55">A31</font></span></a>, <a class="txt" href="#E7:26_3_2_1_1"><span class="lab"><font color="Green" title="E51">A27</font></span></a>, <a class="txt" href="#E26:26_3_2_1_1"><span class="lab"><font color="Green" title="E61">A80</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E29:26_3_2_1_1"/><span class="lab"><font color="Green" title="E62">A81</font></span>: 
<font color="Maroon" title="c12">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E30:26_3_2_1_1"/><span class="lab"><font color="Green" title="E63">A82</font></span>: 
 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="c10">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#K8" title="RELSET_1:func.8">"</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_1_6"/><span class="lab"><font color="Green" title="E64">A83</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_1_6"/>
<font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_1_6"/><span class="lab"><font color="Green" title="E65">A84</font></span>: 
<font color="Maroon" title="c21">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_1_6"/><span class="lab"><font color="Green" title="E66">A85</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E64">A83</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="E7:26_3_2_1_1_6"/><span class="lab"><font color="Green" title="E67">A86</font></span>: 
<font color="Maroon" title="c22">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E65">A84</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A87</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_1"><span class="lab"><font color="Green" title="E53">A29</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_1_6"/><span class="lab"><font color="Green" title="E69">A88</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_1"><span class="lab"><font color="Green" title="E52">A28</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A89</font></span>: <font color="Maroon" title="c12">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E66">A85</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:26_3_2_1_1"><span class="lab"><font color="Green" title="E61">A80</font></span></a>, <a class="txt" href="#E8:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E68">A87</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_1_6"/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:26_3_2_1_1"><span class="lab"><font color="Green" title="E62">A81</font></span></a>, <a class="txt" href="#E9:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E69">A88</font></span></a>, <a class="txt" href="#E7:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E67">A86</font></span></a>, <a class="txt" href="#E10:26_3_2_1_1_6"><span class="lab"><font color="Green" title="E70">A89</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E31:26_3_2_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E32:26_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A90</font></span>: 
<font color="Maroon" title="c19">g2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:26_3_2_1_1"><span class="lab"><font color="Green" title="E63">A82</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E33:26_3_2_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">q9</font>,<font color="Maroon" title="c15">q29</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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">q9</font>,<font color="Maroon" title="c15">q29</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E32:26_3_2_1_1"><span class="lab"><font color="Green" title="E64">A90</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><a NAME="E35:26_3_2_1_1"/><span class="lab"><font color="Green" title="E65">A91</font></span>: 
<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">q29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26_3_2_1_1"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="txt" href="#E11:26_3_2_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E32:26_3_2_1_1"><span class="lab"><font color="Green" title="E64">A90</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E36:26_3_2_1_1"/><span class="lab"><font color="Green" title="E66">A92</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26_3_2_1_1"><span class="lab"><font color="Green" title="E59">A55</font></span></a>, <a class="txt" href="#E11:26_3_2_1_1"><span class="lab"><font color="Green" title="E54">A30</font></span></a>, <a class="txt" href="#E32:26_3_2_1_1"><span class="lab"><font color="Green" title="E64">A90</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E37:26_3_2_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A93</font></span>: 
<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="c18">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">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:26_3_2_1_1"><span class="lab"><font color="Green" title="E65">A91</font></span></a>, <a class="txt" href="#E24:26_3_2_1_1"><span class="lab"><font color="Green" title="E60">A76</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E38:26_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="c18">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="c18">G3</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:26_3_2_1_1"><span class="lab"><font color="Green" title="E66">A92</font></span></a>, <a class="txt" href="#E35:26_3_2_1_1"><span class="lab"><font color="Green" title="E65">A91</font></span></a>, <a class="txt" href="#E24:26_3_2_1_1"><span class="lab"><font color="Green" title="E60">A76</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><span class="kw">hence </span><a NAME="E39:26_3_2_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E9:26"><span class="lab"><font color="Green" title="E34">A8</font></span></a>, <a class="txt" href="#E16:26_3_2_1_1"><span class="lab"><font color="Green" title="E58">A34</font></span></a>, <a class="txt" href="#E37:26_3_2_1_1"><span class="lab"><font color="Green" title="E67">A93</font></span></a>, <a class="ref" href="jordan20.html#T18" target="_self" title="JORDAN20:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_2"/><span class="lab"><font color="Green" title="E48">A94</font></span>: 
<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>)
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><div class="add"><a NAME="E2:26_3_2_1_2"/><span class="lab"><font color="Green" title="E49">A95</font></span>: 
(  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> &amp;  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> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E3:26_3_2_1_2"/>
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26"><span class="lab"><font color="Green" title="E31">A5</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c10">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> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E3:26_3"><span class="lab"><font color="Green" title="E40">A16</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/><a NAME="E5:26_3_2_1_2"/><span class="lab"><font color="Green" title="E50">A96</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c11">Qa9</font> = <font color="Maroon" title="c10">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/><a NAME="E7:26_3_2_1_2"/><span class="lab"><font color="Green" title="E51">A97</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><a NAME="E8:26_3_2_1_2"/><span class="lab"><font color="Green" title="E52">A98</font></span>: 
 <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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E13:26_3"><span class="lab"><font color="Green" title="E46">A22</font></span></a>, <a class="ref" href="jordan16.html#T24" title="JORDAN16:th.24">JORDAN16:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c12">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><span class="kw"> such that </span><br/><a NAME="E10:26_3_2_1_2"/><span class="lab"><font color="Green" title="E53">A99</font></span>: 
<font color="Maroon" title="c12">f2</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E11:26_3_2_1_2"/><span class="lab"><font color="Green" title="E54">A100</font></span>: 
( <font color="Maroon" title="c12">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q1</font> &amp; <font color="Maroon" title="c12">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E55">A101</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c12">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_2"><span class="lab"><font color="Green" title="E53">A99</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E13:26_3_2_1_2"/><span class="lab"><font color="Green" title="E56">A102</font></span>: 
 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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E7:26_3"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E14:26_3_2_1_2"/>
<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> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2"><span class="lab"><font color="Green" title="E48">A94</font></span></a>, <a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E15:26_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E57">A103</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p3</font> &amp;  <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> )
 ;<br/><div><span class="lab"><font color="Green" title="E58">A104</font></span>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_2"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_3_2_1_2_2"/><span class="lab"><font color="Green" title="E59">A105</font></span>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_3_2_1_2_2"/><span class="kw">then </span>
<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 class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E15:26_3_2_1_2"><span class="lab"><font color="Green" title="E57">A103</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_3_2_1_2_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E1:26_3_2_1_2_2"><span class="lab"><font color="Green" title="E59">A105</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><a NAME="E17:26_3_2_1_2"/><span class="kw">then </span>
 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>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">p19</font> = <font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c14">q9</font> = <font color="Maroon" title="c9">q</font>, <font color="Maroon" title="c15">q19</font> = <font color="Maroon" title="c5">q1</font>, <font color="Maroon" title="c16">p29</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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="txt" href="#E12:26"><span class="lab"><font color="Green" title="E37">A13</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="txt" href="#E5:26_3_2_1_2"><span class="lab"><font color="Green" title="E50">A96</font></span></a>, <a class="txt" href="#E2:26_3_2_1_2"><span class="lab"><font color="Green" title="E49">A95</font></span></a>, <a class="txt" href="#E13:26_3_2_1_2"><span class="lab"><font color="Green" title="E56">A102</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_3"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_3_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_3_2_1_2_3_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_3_1_1"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_2_3_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">q9</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q9</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">p29</font> )</span><br/><div class="add"><a NAME="E2:26_3_2_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A106</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="jordan20.html#T21" target="_self" title="JORDAN20:th.21">Th22</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E60">A107</font></span>: 
<font color="Maroon" title="c17">f1</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E61">A108</font></span>: 
( <font color="Maroon" title="c17">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E62">A109</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E60">A107</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E7:26_3_2_1_2_3_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_3_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c18">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_2_3_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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A110</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c18">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<a NAME="E3:26_3_2_1_2_3_1_1_2"/><span class="kw">then </span>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E7:26_3"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E15:26_3_2_1_2"><span class="lab"><font color="Green" title="E57">A103</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>;<br/>
<a NAME="E4:26_3_2_1_2_3_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A111</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_2_3_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="c1">P</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E63">A110</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_2_3_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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_3_1_1_2"><span class="lab"><font color="Green" title="E64">A111</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:26_3_2_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A112</font></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="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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E9:26_3_2_1_2_3_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E4:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E60">A107</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2"><span class="lab"><font color="Green" title="E50">A96</font></span></a>, <a class="txt" href="#E6:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E62">A109</font></span></a>, <a class="txt" href="#E8:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E63">A112</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E11:26_3_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E64">A113</font></span>: 
<font color="Maroon" title="c17">f1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E60">A107</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E12:26_3_2_1_2_3_1_1"/><span class="lab"><font color="Green" title="E65">A114</font></span>: 
 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="c10">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#K8" title="RELSET_1:func.8">"</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_3_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E66">A115</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_2_3_1_1_3"/>
<font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E67">A116</font></span>: 
<font color="Maroon" title="c20">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E68">A117</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E66">A115</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="E7:26_3_2_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E69">A118</font></span>: 
<font color="Maroon" title="c21">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E67">A116</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A119</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E60">A107</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_2_3_1_1_3"/><span class="lab"><font color="Green" title="E71">A120</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E59">A106</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">A121</font></span>: <font color="Maroon" title="c17">f1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c21">G5</font> = 
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E68">A117</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E63">A112</font></span></a>, <a class="txt" href="#E8:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E70">A119</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_2_3_1_1_3"/>
<font color="Maroon" title="c18">g1</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c19">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E64">A113</font></span></a>, <a class="txt" href="#E9:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E71">A120</font></span></a>, <a class="txt" href="#E7:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E69">A118</font></span></a>, <a class="txt" href="#E10:26_3_2_1_2_3_1_1_3"><span class="lab"><font color="Green" title="E72">A121</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E13:26_3_2_1_2_3_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E14:26_3_2_1_2_3_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E66">A122</font></span>: 
<font color="Maroon" title="c18">g1</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E65">A114</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E15:26_3_2_1_2_3_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c14">q9</font>,<font color="Maroon" title="c16">p29</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E61">A108</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><a NAME="E16:26_3_2_1_2_3_1_1"/><span class="kw">then </span>
<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">q9</font>,<font color="Maroon" title="c16">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E61">A108</font></span></a>, <a class="txt" href="#E14:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E66">A122</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E17:26_3_2_1_2_3_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">q9</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q9</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">p29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E61">A108</font></span></a>, <a class="txt" href="#E14:26_3_2_1_2_3_1_1"><span class="lab"><font color="Green" title="E66">A122</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_3_1_2"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E59">A123</font></span>: 
<font color="Maroon" title="c9">q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">q9</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q9</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">p29</font> )</span><br/><div class="add"><span class="kw">consider </span><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="c10">Qa</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_3_2_1_2_3_1_2"/><span class="lab"><font color="Green" title="E60">A124</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">p29</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">p29</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#T3" title="BORSUK_2:th.3">BORSUK_2:3</a></span>;<br/><a NAME="E4:26_3_2_1_2_3_1_2"/>
<font color="Maroon" title="c16">p29</font>,<font color="Maroon" title="c16">p29</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_3_1_2"/><span class="kw">then </span>
<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">p29</font>,<font color="Maroon" title="c16">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_3_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E60">A124</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_3_2_1_2_3_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">q9</font>,<font color="Maroon" title="c16">p29</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q9</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">p29</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E59">A123</font></span></a>, <a class="txt" href="#E3:26_3_2_1_2_3_1_2"><span class="lab"><font color="Green" title="E60">A124</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><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">q9</font>,<font color="Maroon" title="c16">p29</font><span class="kw"> such that </span><br/><a NAME="E21:26_3_2_1_2"/><span class="lab"><font color="Green" title="E59">A125</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">q9</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">p29</font> )
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4"><span class="kw">now </span></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_4_1"><span class="kw">per </span></a><a class="txt" onclick="hs(this)" href="javascript:()"><span class="kw">cases </span></a><span class="hide"><a NAME="E1:26_3_2_1_2_4_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_4_1_1"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_2_4_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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">p19</font>,<font color="Maroon" title="c15">q19</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q19</font> )</span><br/><div class="add"><a NAME="E2:26_3_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A126</font></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="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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E10:26"><span class="lab"><font color="Green" title="E35">A10</font></span></a>, <a class="ref" href="jordan16.html#T24" title="JORDAN16:th.24">JORDAN16:24</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E61">A127</font></span>: 
<font color="Maroon" title="c18">f3</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E62">A128</font></span>: 
( <font color="Maroon" title="c18">f3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c18">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> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal1.html#D1" title="TOPREAL1:def.1">TOPREAL1:def 1</a></span>;<br/><span class="lab"><font color="Green" title="E63">A129</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c18">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E61">A127</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/><a NAME="E7:26_3_2_1_2_4_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_4_1_1_2"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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="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>  or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_2_4_1_1_2"/>
<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="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 class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_2_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A130</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<a NAME="E3:26_3_2_1_2_4_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c6">q2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E3:26"><span class="lab"><font color="Green" title="E29">A3</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a>, <a class="ref" href="jordan6.html#T37" title="JORDAN6:th.37">JORDAN6:37</a></span>;<br/>
<a NAME="E4:26_3_2_1_2_4_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A131</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_2_4_1_1_2"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_2_4_1_1_2"><span class="lab"><font color="Green" title="E64">A130</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_2_4_1_1_2"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_4_1_1_2"><span class="lab"><font color="Green" title="E65">A131</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E8:26_3_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A132</font></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="c5">q1</font>) <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E9:26_3_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E65">A133</font></span>: 
 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="c10">Qa</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/><a NAME="E10:26_3_2_1_2_4_1_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c18">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E4:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E61">A127</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">g3</font> = <font color="Maroon" title="c18">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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E6:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E63">A129</font></span></a>, <a class="txt" href="#E9:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E65">A133</font></span></a>, <a class="txt" href="#E8:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E12:26_3_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E66">A134</font></span>: 
<font color="Maroon" title="c18">f3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E61">A127</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E13:26_3_2_1_2_4_1_1"/><span class="lab"><font color="Green" title="E67">A135</font></span>: 
 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="c10">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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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_4_1_1_3"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E68">A136</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_2_4_1_1_3"/>
<font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E69">A137</font></span>: 
<font color="Maroon" title="c21">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E70">A138</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E68">A136</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="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="E7:26_3_2_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E71">A139</font></span>: 
<font color="Maroon" title="c22">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E69">A137</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E72">A140</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E61">A127</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_2_4_1_1_3"/><span class="lab"><font color="Green" title="E73">A141</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E60">A126</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E74">A142</font></span>: <font color="Maroon" title="c18">f3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="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 class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E70">A138</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c19">g3</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E64">A132</font></span></a>, <a class="txt" href="#E8:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E72">A140</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_2_4_1_1_3"/>
<font color="Maroon" title="c19">g3</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E66">A134</font></span></a>, <a class="txt" href="#E9:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E73">A141</font></span></a>, <a class="txt" href="#E7:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E71">A139</font></span></a>, <a class="txt" href="#E10:26_3_2_1_2_4_1_1_3"><span class="lab"><font color="Green" title="E74">A142</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E14:26_3_2_1_2_4_1_1"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E15:26_3_2_1_2_4_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E68">A143</font></span>: 
<font color="Maroon" title="c19">g3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E67">A135</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E16:26_3_2_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c13">p19</font>,<font color="Maroon" title="c15">q19</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E62">A128</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><a NAME="E17:26_3_2_1_2_4_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c19">g3</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p19</font>,<font color="Maroon" title="c15">q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E62">A128</font></span></a>, <a class="txt" href="#E15:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E68">A143</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E18:26_3_2_1_2_4_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">p19</font>,<font color="Maroon" title="c15">q19</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E62">A128</font></span></a>, <a class="txt" href="#E15:26_3_2_1_2_4_1_1"><span class="lab"><font color="Green" title="E68">A143</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="26_3_2_1_2_4_1_2"><span class="kw">case </span></a><a NAME="E1:26_3_2_1_2_4_1_2"/><span class="lab"><font color="Green" title="E60">A144</font></span>: 
<font color="Maroon" title="c5">q1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  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">p19</font>,<font color="Maroon" title="c15">q19</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q19</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c18">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="c10">Qa</font></span>)</span><span class="kw"> such that </span><br/><a NAME="E3:26_3_2_1_2_4_1_2"/><span class="lab"><font color="Green" title="E61">A145</font></span>: 
( <font color="Maroon" title="c18">g01</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c18">g01</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q19</font> &amp; <font color="Maroon" title="c18">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">q19</font> )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_2.html#T3" title="BORSUK_2:th.3">BORSUK_2:3</a></span>;<br/><a NAME="E4:26_3_2_1_2_4_1_2"/>
<font color="Maroon" title="c15">q19</font>,<font color="Maroon" title="c15">q19</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 ;<br/><a NAME="E5:26_3_2_1_2_4_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c18">g01</font> is    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c15">q19</font>,<font color="Maroon" title="c15">q19</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:26_3_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E61">A145</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><span class="kw">hence </span><a NAME="E6:26_3_2_1_2_4_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">p19</font>,<font color="Maroon" title="c15">q19</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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</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">q19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E60">A144</font></span></a>, <a class="txt" href="#E3:26_3_2_1_2_4_1_2"><span class="lab"><font color="Green" title="E61">A145</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">G3</font> being    <a href="borsuk_2.html#M1" title="BORSUK_2:mode.1">Path</a> of <font color="Maroon" title="c13">p19</font>,<font color="Maroon" title="c15">q19</font><span class="kw"> such that </span><br/><a NAME="E24:26_3_2_1_2"/><span class="lab"><font color="Green" title="E60">A146</font></span>: 
( <font color="Maroon" title="c18">G3</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>  &amp; <font color="Maroon" title="c18">G3</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</font> &amp; <font color="Maroon" title="c18">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">q19</font> )
 ;<br/><a NAME="E25:26_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="c10">Qa</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_5"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c19">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <span class="comment"><font color="firebrick">:: according to </font></span><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not <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>  or <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font> )</span><br/>

<span class="kw">assume </span><a NAME="E1:26_3_2_1_2_5"/>
<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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font></span><br/>

<a NAME="E2:26_3_2_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A147</font></span>: 
 ex <font color="Olive" title="b1">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> st <br/>( <font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">p</font> &amp;  <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> )
 
;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="26_3_2_1_2_5_1"><span class="kw">now </span></a><div class="add"><span class="kw">assume </span><a NAME="E1:26_3_2_1_2_5_1"/><span class="lab"><font color="Green" title="E62">A148</font></span>: 
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">q2</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> contradiction</span><br/><a NAME="E2:26_3_2_1_2_5_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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E10:26_3"><span class="lab"><font color="Green" title="E44">A20</font></span></a>, <a class="ref" href="jordan5c.html#T10" title="JORDAN5C:th.10">JORDAN5C:10</a></span>;<br/><span class="kw">hence </span><a NAME="E3:26_3_2_1_2_5_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E11:26_3"><span class="lab"><font color="Green" title="E45">A21</font></span></a>, <a class="txt" href="#E7:26_3"><span class="lab"><font color="Green" title="E43">A19</font></span></a>, <a class="txt" href="#E2:26_3_2_1_2_5"><span class="lab"><font color="Green" title="E61">A147</font></span></a>, <a class="txt" href="#E1:26_3_2_1_2_5_1"><span class="lab"><font color="Green" title="E62">A148</font></span></a>, <a class="ref" href="jordan5c.html#T12" title="JORDAN5C:th.12">JORDAN5C:12</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<a NAME="E4:26_3_2_1_2_5"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A149</font></span>: 
 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>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E5:26_3_2_1_2_5"/>
<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>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26_3_2_1_2_5"><span class="lab"><font color="Green" title="E61">A147</font></span></a>, <a class="ref" href="jordan5c.html#D3" title="JORDAN5C:def.3">JORDAN5C:def 3</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:26_3_2_1_2_5"/>
<font color="Maroon" title="c19">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26"><span class="lab"><font color="Green" title="E30">A4</font></span></a>, <a class="txt" href="#E4:26_3_2_1_2_5"><span class="lab"><font color="Green" title="E62">A149</font></span></a>, <a class="ref" href="xboole_0.html#D5" title="XBOOLE_0:def.5">XBOOLE_0:def 5</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E26:26_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A150</font></span>: 
 <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="c10">Qa</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="jordan6.html#T26" title="JORDAN6:th.26">JORDAN6:26</a></span>;<br/><a NAME="E27:26_3_2_1_2"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c12">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>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3"><span class="lab"><font color="Green" title="E42">A18</font></span></a>, <a class="txt" href="#E10:26_3_2_1_2"><span class="lab"><font color="Green" title="E53">A99</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c19">g2</font> = <font color="Maroon" title="c12">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="c10">Qa</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E12:26_3_2_1_2"><span class="lab"><font color="Green" title="E55">A101</font></span></a>, <a class="txt" href="#E7:26_3_2_1_2"><span class="lab"><font color="Green" title="E51">A97</font></span></a>, <a class="txt" href="#E26:26_3_2_1_2"><span class="lab"><font color="Green" title="E61">A150</font></span></a>, <a class="ref" href="funct_2.html#T2" title="FUNCT_2:th.2">FUNCT_2:2</a></span>;<br/><a NAME="E29:26_3_2_1_2"/><span class="lab"><font color="Green" title="E62">A151</font></span>: 
<font color="Maroon" title="c12">f2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_2"><span class="lab"><font color="Green" title="E53">A99</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>;<br/><a NAME="E30:26_3_2_1_2"/><span class="lab"><font color="Green" title="E63">A152</font></span>: 
 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="c10">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#K8" title="RELSET_1:func.8">"</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"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><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="c10">Qa</font></span>)</span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  implies <font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>  )</span><br/>

<a NAME="E1:26_3_2_1_2_6"/><span class="lab"><font color="Green" title="E64">A153</font></span>: 
<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="c10">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="c11">Qa9</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T7" title="PRE_TOPC:th.7">PRE_TOPC:7</a>, <a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">assume </span><a NAME="E2:26_3_2_1_2_6"/>
<font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> </span><br/>

<span class="kw">then </span><span class="kw">consider </span><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><span class="kw"> such that </span><br/><a NAME="E4:26_3_2_1_2_6"/><span class="lab"><font color="Green" title="E65">A154</font></span>: 
<font color="Maroon" title="c21">G4</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">and </span><br/><a NAME="E5:26_3_2_1_2_6"/><span class="lab"><font color="Green" title="E66">A155</font></span>: 
<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="c11">Qa9</font></span>)</span></span>)</span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E64">A153</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="kw">reconsider </span><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="E7:26_3_2_1_2_6"/><span class="lab"><font color="Green" title="E67">A156</font></span>: 
<font color="Maroon" title="c22">G5</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E65">A154</font></span></a>, <a class="ref" href="tops_2.html#T24" title="TOPS_2:th.24">TOPS_2:24</a></span>;<br/>
<span class="lab"><font color="Green" title="E68">A157</font></span>:  <a href="relset_1.html#K2" title="RELSET_1:func.2">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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:26_3_2_1_2"><span class="lab"><font color="Green" title="E53">A99</font></span></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></span>
<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>)
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
;<br/>
<a NAME="E9:26_3_2_1_2_6"/><span class="lab"><font color="Green" title="E69">A158</font></span>: 
<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>)
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:26_3_2_1_2"><span class="lab"><font color="Green" title="E52">A98</font></span></a>, <a class="ref" href="topreal1.html#T1" title="TOPREAL1:th.1">TOPREAL1:1</a></span>;<br/>
<span class="lab"><font color="Green" title="E70">A159</font></span>: <font color="Maroon" title="c12">f2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c22">G5</font> = 
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
;<br/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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="c11">Qa9</font></span>)</span></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E66">A155</font></span></a>, <a class="ref" href="funct_1.html#T68" title="FUNCT_1:th.68">FUNCT_1:68</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c11">Qa9</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#D5" title="PRE_TOPC:def.5">PRE_TOPC:def 5</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K2" title="RELSET_1:func.2">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="c11">Qa9</font></span>)</span></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T133" title="RELAT_1:th.133">RELAT_1:133</a></span>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</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#K8" title="RELSET_1:func.8">"</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>
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E26:26_3_2_1_2"><span class="lab"><font color="Green" title="E61">A150</font></span></a>, <a class="txt" href="#E8:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E68">A157</font></span></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E12:26_3_2_1_2_6"/>
<font color="Maroon" title="c19">g2</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c20">G</font> is  <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:26_3_2_1_2"><span class="lab"><font color="Green" title="E62">A151</font></span></a>, <a class="txt" href="#E9:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E69">A158</font></span></a>, <a class="txt" href="#E7:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E67">A156</font></span></a>, <a class="txt" href="#E10:26_3_2_1_2_6"><span class="lab"><font color="Green" title="E70">A159</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div><span class="kw">end;</span></div><a NAME="E31:26_3_2_1_2"/>
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</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="c10">Qa</font></span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><a NAME="E32:26_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A160</font></span>: 
<font color="Maroon" title="c19">g2</font> is  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:26_3_2_1_2"><span class="lab"><font color="Green" title="E63">A152</font></span></a>, <a class="ref" href="tops_2.html#T43" title="TOPS_2:th.43">TOPS_2:43</a></span>;<br/><a NAME="E33:26_3_2_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c15">q19</font>,<font color="Maroon" title="c14">q9</font> <a href="borsuk_2.html#R2" title="BORSUK_2:pred.2">are_connected</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_2"><span class="lab"><font color="Green" title="E54">A100</font></span></a>, <a class="ref" href="borsuk_2.html#D1" title="BORSUK_2:def.1">BORSUK_2:def 1</a></span>;<br/><span class="kw">then </span><span class="kw">reconsider </span><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">q19</font>,<font color="Maroon" title="c14">q9</font> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E11:26_3_2_1_2"><span class="lab"><font color="Green" title="E54">A100</font></span></a>, <a class="txt" href="#E32:26_3_2_1_2"><span class="lab"><font color="Green" title="E64">A160</font></span></a>, <a class="ref" href="borsuk_2.html#D2" title="BORSUK_2:def.2">BORSUK_2:def 2</a></span>;<br/><a NAME="E35:26_3_2_1_2"/><span class="lab"><font color="Green" title="E65">A161</font></span>: 
<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">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26_3_2_1_2"><span class="lab"><font color="Green" title="E59">A125</font></span></a>, <a class="txt" href="#E11:26_3_2_1_2"><span class="lab"><font color="Green" title="E54">A100</font></span></a>, <a class="txt" href="#E32:26_3_2_1_2"><span class="lab"><font color="Green" title="E64">A160</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E36:26_3_2_1_2"/><span class="lab"><font color="Green" title="E66">A162</font></span>: 
( <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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">q19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:26_3_2_1_2"><span class="lab"><font color="Green" title="E59">A125</font></span></a>, <a class="txt" href="#E11:26_3_2_1_2"><span class="lab"><font color="Green" title="E54">A100</font></span></a>, <a class="txt" href="#E32:26_3_2_1_2"><span class="lab"><font color="Green" title="E64">A160</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E37:26_3_2_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E67">A163</font></span>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c18">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">p29</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E35:26_3_2_1_2"><span class="lab"><font color="Green" title="E65">A161</font></span></a>, <a class="txt" href="#E24:26_3_2_1_2"><span class="lab"><font color="Green" title="E60">A146</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><a NAME="E38:26_3_2_1_2"/>
( <font color="Maroon" title="c18">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="c18">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> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">p19</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E36:26_3_2_1_2"><span class="lab"><font color="Green" title="E66">A162</font></span></a>, <a class="txt" href="#E35:26_3_2_1_2"><span class="lab"><font color="Green" title="E65">A161</font></span></a>, <a class="txt" href="#E24:26_3_2_1_2"><span class="lab"><font color="Green" title="E60">A146</font></span></a>, <a class="ref" href="borsuk_2.html#T14" title="BORSUK_2:th.14">BORSUK_2:14</a></span>;<br/><span class="kw">hence </span><a NAME="E39:26_3_2_1_2"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:26"><span class="lab"><font color="Green" title="E27">A1</font></span></a>, <a class="txt" href="#E8:26"><span class="lab"><font color="Green" title="E33">A7</font></span></a>, <a class="txt" href="#E11:26"><span class="lab"><font color="Green" title="E36">A11</font></span></a>, <a class="txt" href="#E16:26_3_2_1_2"><span class="lab"><font color="Green" title="E58">A104</font></span></a>, <a class="txt" href="#E37:26_3_2_1_2"><span class="lab"><font color="Green" title="E67">A163</font></span></a>, <a class="ref" href="jordan20.html#T18" target="_self" title="JORDAN20:th.18">Th18</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div></div><span class="kw">end;</span></div><span class="kw">hence </span><a NAME="E18:26_3"/>
contradiction
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><a NAME="E14:26"/>
<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>)
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:26"><span class="lab"><font color="Green" title="E28">A2</font></span></a>, <a class="txt" href="#E7:26"><span class="lab"><font color="Green" title="E32">A6</font></span></a>, <a class="ref" href="jordan20.html#T20" target="_self" title="JORDAN20:th.20">Th21</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


</div>
