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

<span class="kw">let </span><font color="Maroon" title="c1">D</font> be   <a href="topreal2.html#NM1" title="TOPREAL2:NM.1">Simple_closed_curve</a>; <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">C</font> being   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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>  st <font color="Olive" title="b1">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</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> ex <font color="Olive" title="b4">B</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> st <br/>( <font color="Olive" title="b4">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">let </span><font color="Maroon" title="c2">C</font> be   <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</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"> ( <font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> ) )</span><br/>



<span class="kw">assume </span><a NAME="E1:59"/><span class="lab"><font color="Green" title="E50">A1</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<a NAME="E2:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E51">A2</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/>
<a NAME="E3:59"/><span class="lab"><font color="Green" title="E52">A3</font></span>: 
 for <font color="Olive" title="b1">C</font> being   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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>  st  not <font color="Olive" title="b1">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Olive" title="b1">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font> holds <br/> ex <font color="Olive" title="b2">p1</font>, <font color="Olive" title="b3">p2</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> ex <font color="Olive" title="b4">B</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> st <br/>( <font color="Olive" title="b4">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Olive" title="b2">p1</font>,<font color="Olive" title="b3">p2</font> &amp; <font color="Olive" title="b1">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b4">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">let </span><font color="Maroon" title="c3">C</font> be   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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"> (  not <font color="Maroon" title="c3">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  &amp; <font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font> implies  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> ) )</span><br/>

<span class="kw">assume </span><a NAME="E1:59_1"/>
 not <font color="Maroon" title="c3">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</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="c3">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font> or  ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> ) )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c4">d1</font>, <font color="Maroon" title="c5">d2</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><span class="kw"> such that </span><br/><a NAME="E3:59_1"/><span class="lab"><font color="Green" title="E53">A4</font></span>: 
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E4:59_1"/><span class="lab"><font color="Green" title="E54">A5</font></span>: 
<font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">C</font>
 <span class="kw">and </span><br/><a NAME="E5:59_1"/><span class="lab"><font color="Green" title="E55">A6</font></span>: 
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c5">d2</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="subset_1.html#T45" title="SUBSET_1:th.45">SUBSET_1:45</a>, <a class="ref" href="zfmisc_1.html#D10" title="ZFMISC_1:def.10">ZFMISC_1:def 10</a></span>;<br/>
<span class="kw">assume </span><a NAME="E6:59_1"/>
<font color="Maroon" title="c3">C</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c6">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><span class="kw"> such that </span><br/><a NAME="E8:59_1"/><span class="lab"><font color="Green" title="E56">A7</font></span>: 
<font color="Maroon" title="c6">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">and </span><br/><a NAME="E9:59_1"/><span class="lab"><font color="Green" title="E57">A8</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="subset_1.html#T44" title="SUBSET_1:th.44">SUBSET_1:44</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c7">Dp</font> = <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</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"><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="#E3:59_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a></span>;<br/>
<a NAME="E11:59_1"/>
<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="c7">Dp</font>, <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  <a href="borsuk_3.html#R2" title="BORSUK_3:pred.2">are_homeomorphic</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E8:59_1"><span class="lab"><font color="Green" title="E56">A7</font></span></a>, <a class="ref" href="borsuk_4.html#T52" target="_self" title="BORSUK_4:th.52">Th77</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="c7">Dp</font></span>)</span>,<a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a><span class="kw"> such that </span><br/><a NAME="E13:59_1"/><span class="lab"><font color="Green" title="E58">A9</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="tops_2.html#V3" title="TOPS_2:attr.3">being_homeomorphism</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="t_0topsp.html#D1" title="T_0TOPSP:def.1">T_0TOPSP:def 1</a></span>;<br/>
<a NAME="E14:59_1"/>
<font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a></span>;<br/>
<a NAME="E15:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c7">Dp</font></span>)</span>
 
<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="E16:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A10</font></span>: 
<font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E17:59_1"/>
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c6">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a></span>;<br/>
<a NAME="E18:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <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="c7">Dp</font></span>)</span>
 
<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="E19:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A11</font></span>: 
<font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E20:59_1"/><span class="lab"><font color="Green" title="E61">A12</font></span>: 
<font color="Maroon" title="c8">f</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_1"><span class="lab"><font color="Green" title="E58">A9</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="E21:59_1"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">Dp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E22:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E62">A13</font></span>: 
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E23:59_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</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 <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="c7">Dp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E24:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E63">A14</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</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="ref" href="borsuk_1.html#T1" title="BORSUK_1:th.1">BORSUK_1:1</a></span>;<br/>
<a NAME="E25:59_1"/>
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</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 <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="c7">Dp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></span>;<br/>
<a NAME="E26:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E64">A15</font></span>: 
 <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">Dp</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="c9">C9</font> = <font color="Maroon" title="c3">C</font> 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> <font color="Maroon" title="c7">Dp</font></span>)</span> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E28:59_1"/>
<font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <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="c7">Dp</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_1"><span class="lab"><font color="Green" title="E57">A8</font></span></a>, <a class="ref" href="pre_topc.html#T8" title="PRE_TOPC:th.8">PRE_TOPC:8</a></span>;<br/>
<a NAME="E29:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E65">A16</font></span>: 
<font color="Maroon" title="c9">C9</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T2" title="COMPTS_1:th.2">COMPTS_1:2</a></span>;<br/>
<span class="kw">set </span><font color="Maroon" title="c10">fC</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C9</font>;<br/>
<a NAME="E30:59_1"/><span class="lab"><font color="Green" title="E66">A17</font></span>: 
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_1"><span class="lab"><font color="Green" title="E58">A9</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="E31:59_1"/>
<font color="Maroon" title="c8">f</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="#E13:59_1"><span class="lab"><font color="Green" title="E58">A9</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="c11">fC</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C9</font> as   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a> <span class="kw">by</span> <span class="lab"><a class="txt" href="#E29:59_1"><span class="lab"><font color="Green" title="E65">A16</font></span></a>, <a class="txt" href="#E30:59_1"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="ref" href="compts_1.html#T15" title="COMPTS_1:th.15">COMPTS_1:15</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c12">fC9</font> = <font color="Maroon" title="c11">fC</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> <span class="kw">by</span> <span class="lab"><a class="ref" href="pre_topc.html#T11" title="PRE_TOPC:th.11">PRE_TOPC:11</a></span>;<br/>
<a NAME="E34:59_1"/><span class="lab"><font color="Green" title="E67">A18</font></span>: 
<font color="Maroon" title="c12">fC9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>
 
;<br/>
<a NAME="E35:59_1"/><span class="lab"><font color="Green" title="E68">A19</font></span>: 
 for <font color="Olive" title="b1">P</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>  st <font color="Olive" title="b1">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">fC9</font> holds <br/><font color="Olive" title="b1">P</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
;<br/>
<a NAME="E36:59_1"/>
<font color="Maroon" title="c12">fC9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_4.html#K1" title="BORSUK_4:func.1">I(01)</a>
 
;<br/>
<a NAME="E37:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E69">A20</font></span>: 
<font color="Maroon" title="c12">fC9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></span>;<br/>
<a NAME="E38:59_1"/><span class="lab"><font color="Green" title="E70">A21</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_1"><span class="lab"><font color="Green" title="E54">A5</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c13">fC9</font> = <font color="Maroon" title="c12">fC9</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="#E34:59_1"><span class="lab"><font color="Green" title="E67">A18</font></span></a>, <a class="txt" href="#E35:59_1"><span class="lab"><font color="Green" title="E68">A19</font></span></a>, <a class="ref" href="compts_1.html#T2" title="COMPTS_1:th.2">COMPTS_1:2</a></span>;<br/>
<span class="kw">consider </span><font color="Maroon" title="c14">p1</font>, <font color="Maroon" title="c15">p2</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a><span class="kw"> such that </span><br/><a NAME="E41:59_1"/><span class="lab"><font color="Green" title="E71">A22</font></span>: 
<font color="Maroon" title="c14">p1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">p2</font>
 <span class="kw">and </span><br/><a NAME="E42:59_1"/><span class="lab"><font color="Green" title="E72">A23</font></span>: 
<font color="Maroon" title="c13">fC9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <span class="kw">and </span><br/><a NAME="E43:59_1"/><span class="lab"><font color="Green" title="E73">A24</font></span>: 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E37:59_1"><span class="lab"><font color="Green" title="E69">A20</font></span></a>, <a class="ref" href="borsuk_4.html#T58" target="_self" title="BORSUK_4:th.58">Th83</a></span>;<br/>
<span class="kw">reconsider </span><font color="Maroon" title="c16">E</font> = <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><font color="Maroon" title="c14">p1</font>,<font color="Maroon" title="c15">p2</font></span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</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="#E41:59_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="ref" href="borsuk_4.html#T24" target="_self" title="BORSUK_4:th.24">Th49</a></span>;<br/>
<a NAME="E45:59_1"/><span class="lab"><font color="Green" title="E74">A25</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c16">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<a NAME="E46:59_1"/><span class="lab"><font color="Green" title="E75">A26</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <font color="Maroon" title="c9">C9</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_1"><span class="lab"><font color="Green" title="E53">A4</font></span></a>, <a class="ref" href="funct_2.html#T35" title="FUNCT_2:th.35">FUNCT_2:35</a></span>;<br/>
<a NAME="E47:59_1"/>
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c15">p2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_1_1"/>
<font color="Maroon" title="c14">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">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="E2:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A27</font></span>: 
<font color="Maroon" title="c13">fC9</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="seq_4.html#K1" title="SEQ_4:func.1">{</a><span class="default"><font color="Maroon" title="c14">p1</font></span><a href="seq_4.html#K1" title="SEQ_4:func.1">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:59_1"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="ref" href="xxreal_1.html#T17" title="XXREAL_1:th.17">XXREAL_1:17</a></span>;<br/>
<a NAME="E3:59_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E77">A28</font></span>: 
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">d2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E38:59_1"><span class="lab"><font color="Green" title="E70">A21</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:59_1_1"/>
<font color="Maroon" title="c8">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">p1</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E46:59_1"><span class="lab"><font color="Green" title="E75">A26</font></span></a>, <a class="txt" href="#E2:59_1_1"><span class="lab"><font color="Green" title="E76">A27</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:59_1_1"/>
contradiction
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E5:59_1"><span class="lab"><font color="Green" title="E55">A6</font></span></a>, <a class="txt" href="#E19:59_1"><span class="lab"><font color="Green" title="E60">A11</font></span></a>, <a class="txt" href="#E16:59_1"><span class="lab"><font color="Green" title="E59">A10</font></span></a>, <a class="txt" href="#E20:59_1"><span class="lab"><font color="Green" title="E61">A12</font></span></a>, <a class="txt" href="#E3:59_1_1"><span class="lab"><font color="Green" title="E77">A28</font></span></a>, <a class="ref" href="funct_1.html#D4" title="FUNCT_1:def.4">FUNCT_1:def 4</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="E48:59_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E76">A29</font></span>: 
<font color="Maroon" title="c14">p1</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c15">p2</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E41:59_1"><span class="lab"><font color="Green" title="E71">A22</font></span></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></span>;<br/>
<a NAME="E49:59_1"/>
<font color="Maroon" title="c16">E</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:59_1"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="txt" href="#E43:59_1"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></span>;<br/>
<span class="kw">then </span><span class="kw">reconsider </span><font color="Maroon" title="c17">B</font> = <font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c16">E</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="#E45:59_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E24:59_1"><span class="lab"><font color="Green" title="E63">A14</font></span></a>, <a class="ref" href="relat_1.html#T139" title="RELAT_1:th.139">RELAT_1:139</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></span>;<br/>
<a NAME="E51:59_1"/>
 <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Maroon" title="c8">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">].</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a>,1</span><a href="rcomp_1.html#K2" title="RCOMP_1:func.2">.[</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E30:59_1"><span class="lab"><font color="Green" title="E66">A17</font></span></a>, <a class="ref" href="borsuk_4.html#D1" target="_self" title="BORSUK_4:def.1">Def1</a></span>;<br/>
<a NAME="E52:59_1"/><span class="kw">then </span>
<font color="Maroon" title="c8">f</font> <a href="relset_1.html#K7" title="RELSET_1:func.7">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">f</font> <a href="relset_1.html#K8" title="RELSET_1:func.8">"</a> <font color="Maroon" title="c16">E</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">E</font>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E43:59_1"><span class="lab"><font color="Green" title="E73">A24</font></span></a>, <a class="ref" href="funct_1.html#T77" title="FUNCT_1:th.77">FUNCT_1:77</a></span>;<br/>
<span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c18">s1</font>, <font color="Maroon" title="c19">s2</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><span class="kw"> such that </span><br/><a NAME="E54:59_1"/><span class="lab"><font color="Green" title="E77">A30</font></span>: 
<font color="Maroon" title="c17">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_1"><span class="lab"><font color="Green" title="E58">A9</font></span></a>, <a class="txt" href="#E48:59_1"><span class="lab"><font color="Green" title="E76">A29</font></span></a>, <a class="txt" href="#E26:59_1"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="ref" href="borsuk_4.html#T55" target="_self" title="BORSUK_4:th.55">Th80</a>, <a class="ref" href="relat_1.html#T132" title="RELAT_1:th.132">RELAT_1:132</a></span>;<br/>
<span class="kw">take </span>
<font color="Maroon" title="c18">s1</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">p2</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> ex <font color="Olive" title="b2">B</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> st <br/>( <font color="Olive" title="b2">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Olive" title="b1">p2</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c19">s2</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">B</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> st <br/>( <font color="Olive" title="b1">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<span class="kw">take </span>
<font color="Maroon" title="c17">B</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="c17">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font> &amp; <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<span class="kw">thus </span><a NAME="E55:59_1"/>
<font color="Maroon" title="c17">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">s1</font>,<font color="Maroon" title="c19">s2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E54:59_1"><span class="lab"><font color="Green" title="E77">A30</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"> ( <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/>

<a NAME="E56:59_1"/>
<font color="Maroon" title="c7">Dp</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T36" title="XBOOLE_1:th.36">XBOOLE_1:36</a></span>;<br/>
<span class="kw">hence </span><a NAME="E57:59_1"/>
( <font color="Maroon" title="c3">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">B</font> &amp; <font color="Maroon" title="c17">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E42:59_1"><span class="lab"><font color="Green" title="E72">A23</font></span></a>, <a class="txt" href="#E45:59_1"><span class="lab"><font color="Green" title="E74">A25</font></span></a>, <a class="txt" href="#E26:59_1"><span class="lab"><font color="Green" title="E64">A15</font></span></a>, <a class="txt" href="#E22:59_1"><span class="lab"><font color="Green" title="E62">A13</font></span></a>, <a class="ref" href="funct_1.html#T93" title="FUNCT_1:th.93">FUNCT_1:93</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</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="E4:59"/>
<font color="Maroon" title="c1">D</font> is  <a href="jordan2c.html#V1" title="JORDAN2C:attr.1">Bounded</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="jordan2c.html#T65" title="JORDAN2C:th.65">JORDAN2C:65</a></span>;<br/>
<a NAME="E5:59"/><span class="kw">then </span><span class="lab"><font color="Green" title="E53">A31</font></span>: 
<font color="Maroon" title="c2">C</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="jordan2c.html#T12" title="JORDAN2C:th.12">JORDAN2C:12</a>, <a class="ref" href="topreal6.html#T79" title="TOPREAL6:th.79">TOPREAL6:79</a></span>;<br/>
<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2"><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="E1:59_2"/>
( <font color="Maroon" title="c2">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  or  not <font color="Maroon" title="c2">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_1"/><span class="lab"><font color="Green" title="E54">A32</font></span>: 
<font color="Maroon" title="c2">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_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="E1:59_2_1_1"/>
( <font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  or <font color="Maroon" title="c2">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a>  )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_1_1_1"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_1_1"/><span class="lab"><font color="Green" title="E55">A33</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><div class="add"><span class="kw">consider </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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><span class="kw"> such that </span><br/><a NAME="E3:59_2_1_1_1"/><span class="lab"><font color="Green" title="E56">A34</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">q</font>
 <span class="kw">and </span><br/><a NAME="E4:59_2_1_1_1"/><span class="lab"><font color="Green" title="E57">A35</font></span>: 
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">and </span><br/><a NAME="E5:59_2_1_1_1"/><span class="lab"><font color="Green" title="E58">A36</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="topreal2.html#T4" title="TOPREAL2:th.4">TOPREAL2:4</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">CC</font> = <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c4">q</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span> 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="E7:59_2_1_1_1"/><span class="lab"><font color="Green" title="E59">A37</font></span>: 
<font color="Maroon" title="c4">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E8:59_2_1_1_1"/>
<font color="Maroon" title="c3">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">CC</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a></span>;<br/><a NAME="E9:59_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E60">A38</font></span>: 
 not <font color="Maroon" title="c5">CC</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E7:59_2_1_1_1"><span class="lab"><font color="Green" title="E59">A37</font></span></a>, <a class="ref" href="zfmisc_1.html#D10" title="ZFMISC_1:def.10">ZFMISC_1:def 10</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">pp</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">p</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>, <font color="Maroon" title="c7">qq</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">q</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> 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="E11:59_2_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">pp</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c7">qq</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>;<br/><a NAME="E12:59_2_1_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E61">A39</font></span>: 
<font color="Maroon" title="c5">CC</font> is  <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></span>;<br/><a NAME="E13:59_2_1_1_1"/><span class="lab"><font color="Green" title="E62">A40</font></span>: 
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_1_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_1_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</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="E2:59_2_1_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c5">CC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<a NAME="E3:59_2_1_1_1_1"/><span class="kw">then </span>
 not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</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">D</font></span>)</span> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1_1"><span class="lab"><font color="Green" title="E56">A34</font></span></a>, <a class="txt" href="#E4:59_2_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E5:59_2_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="jordan6.html#T47" title="JORDAN6:th.47">JORDAN6:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E4:59_2_1_1_1_1"/>
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><a NAME="E14:59_2_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E4:59_2_1_1_1"><span class="lab"><font color="Green" title="E57">A35</font></span></a>, <a class="txt" href="#E5:59_2_1_1_1"><span class="lab"><font color="Green" title="E58">A36</font></span></a>, <a class="ref" href="zfmisc_1.html#T32" title="ZFMISC_1:th.32">ZFMISC_1:32</a></span>;<br/><a NAME="E15:59_2_1_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c5">CC</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E13:59_2_1_1_1"><span class="lab"><font color="Green" title="E62">A40</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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>, <font color="Maroon" title="c10">B</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="E17:59_2_1_1_1"/><span class="lab"><font color="Green" title="E63">A41</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <span class="kw">and </span><br/><a NAME="E18:59_2_1_1_1"/>
<font color="Maroon" title="c5">CC</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">and </span><br/><a NAME="E19:59_2_1_1_1"/><span class="lab"><font color="Green" title="E64">A42</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E9:59_2_1_1_1"><span class="lab"><font color="Green" title="E60">A38</font></span></a>, <a class="txt" href="#E12:59_2_1_1_1"><span class="lab"><font color="Green" title="E61">A39</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">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">p2</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> ex <font color="Olive" title="b2">B</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> st <br/>( <font color="Olive" title="b2">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Olive" title="b1">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c9">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">B</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> st <br/>( <font color="Olive" title="b1">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c10">B</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="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">thus </span><a NAME="E20:59_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E17:59_2_1_1_1"><span class="lab"><font color="Green" title="E63">A41</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"> ( <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">thus </span><a NAME="E21:59_2_1_1_1"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1_1_1"><span class="lab"><font color="Green" title="E55">A33</font></span></a>, <a class="ref" href="xboole_1.html#T2" title="XBOOLE_1:th.2">XBOOLE_1:2</a></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="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font></span><br/><span class="kw">thus </span><a NAME="E22:59_2_1_1_1"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_2_1_1_1"><span class="lab"><font color="Green" title="E64">A42</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="59_2_1_1_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_1_1_2"/>
<font color="Maroon" title="c2">C</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ; <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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><div class="add"><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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="E3:59_2_1_1_2"/><span class="lab"><font color="Green" title="E55">A43</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c3">x</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59_2_1"><span class="lab"><font color="Green" title="E54">A32</font></span></a>, <a class="ref" href="borsuk_4.html#T1" target="_self" title="BORSUK_4:th.1">Th2</a></span>;<br/><span class="kw">consider </span><font color="Maroon" title="c4">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c1">D</font><span class="kw"> such that </span><br/><a NAME="E5:59_2_1_1_2"/><span class="lab"><font color="Green" title="E56">A44</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">y</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="borsuk_4.html#T2" target="_self" title="BORSUK_4:th.2">Th3</a></span>;<br/><span class="kw">reconsider </span><font color="Maroon" title="c5">y9</font> = <font color="Maroon" title="c4">y</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c6">Y</font> = <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c5">y9</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/><span class="kw">reconsider </span><font color="Maroon" title="c7">Cy</font> = <font color="Maroon" title="c2">C</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c6">Y</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>   <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</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="#E5:59"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="ref" href="compts_1.html#T10" title="COMPTS_1:th.10">COMPTS_1:10</a></span>;<br/><a NAME="E9:59_2_1_1_2"/><span class="lab"><font color="Green" title="E57">A45</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">C</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1_2"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></span>;<br/><a NAME="E10:59_2_1_1_2"/><span class="lab"><font color="Green" title="E58">A46</font></span>: 
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c1">D</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="59_2_1_1_2_1"><span class="kw">proof </span></a><div class="add">

<span class="kw">assume </span><a NAME="E1:59_2_1_1_2_1"/>
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">D</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="E2:59_2_1_1_2_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A47</font></span>: 
<font color="Maroon" title="c1">D</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></span>;<br/>
<a NAME="E3:59_2_1_1_2_1"/>
<font color="Maroon" title="c7">Cy</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K7" title="DOMAIN_1:func.7">{</a><span class="default"><font color="Maroon" title="c3">x</font>,<font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K7" title="DOMAIN_1:func.7">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59_2_1_1_2"><span class="lab"><font color="Green" title="E55">A43</font></span></a>, <a class="ref" href="enumset1.html#T1" title="ENUMSET1:th.1">ENUMSET1:1</a></span>;<br/>
<a NAME="E4:59_2_1_1_2_1"/><span class="kw">then </span>
 not  <a href="struct_0.html#K1" title="STRUCT_0:func.1">{}</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">D</font></span>)</span> is  <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a> 
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="txt" href="#E9:59_2_1_1_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E5:59_2_1_1_2"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E2:59_2_1_1_2_1"><span class="lab"><font color="Green" title="E59">A47</font></span></a>, <a class="ref" href="jordan6.html#T47" title="JORDAN6:th.47">JORDAN6:47</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:59_2_1_1_2_1"/>
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><a NAME="E11:59_2_1_1_2"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c4">y</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 ;<br/><a NAME="E12:59_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c7">Cy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E2:59"><span class="lab"><font color="Green" title="E51">A2</font></span></a>, <a class="ref" href="xboole_1.html#T8" title="XBOOLE_1:th.8">XBOOLE_1:8</a></span>;<br/><a NAME="E13:59_2_1_1_2"/><span class="kw">then </span><span class="lab"><font color="Green" title="E59">A48</font></span>: 
<font color="Maroon" title="c7">Cy</font> <a href="xboole_0.html#R2" title="XBOOLE_0:pred.2">c&lt;</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E10:59_2_1_1_2"><span class="lab"><font color="Green" title="E58">A46</font></span></a>, <a class="ref" href="xboole_0.html#D8" title="XBOOLE_0:def.8">XBOOLE_0:def 8</a></span>;<br/><a NAME="E14:59_2_1_1_2"/><span class="lab"><font color="Green" title="E60">A49</font></span>: 
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">Cy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></span>;<br/><a NAME="E15:59_2_1_1_2"/>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c6">Y</font>
 <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="E16:59_2_1_1_2"/><span class="kw">then </span>
<font color="Maroon" title="c4">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">Cy</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></span>;<br/><a NAME="E17:59_2_1_1_2"/><span class="kw">then </span>
 not <font color="Maroon" title="c7">Cy</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E9:59_2_1_1_2"><span class="lab"><font color="Green" title="E57">A45</font></span></a>, <a class="txt" href="#E5:59_2_1_1_2"><span class="lab"><font color="Green" title="E56">A44</font></span></a>, <a class="txt" href="#E14:59_2_1_1_2"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="ref" href="zfmisc_1.html#D10" title="ZFMISC_1:def.10">ZFMISC_1:def 10</a></span>;<br/><span class="kw">then </span><span class="kw">consider </span><font color="Maroon" title="c8">p1</font>, <font color="Maroon" title="c9">p2</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>, <font color="Maroon" title="c10">B</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="E19:59_2_1_1_2"/><span class="lab"><font color="Green" title="E61">A50</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <span class="kw">and </span><br/><a NAME="E20:59_2_1_1_2"/><span class="lab"><font color="Green" title="E62">A51</font></span>: 
<font color="Maroon" title="c7">Cy</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">and </span><br/><a NAME="E21:59_2_1_1_2"/><span class="lab"><font color="Green" title="E63">A52</font></span>: 
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E52">A3</font></span></a>, <a class="txt" href="#E13:59_2_1_1_2"><span class="lab"><font color="Green" title="E59">A48</font></span></a></span>;<br/><span class="kw">take </span>
<font color="Maroon" title="c8">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">p2</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> ex <font color="Olive" title="b2">B</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> st <br/>( <font color="Olive" title="b2">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Olive" title="b1">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b2">B</font> &amp; <font color="Olive" title="b2">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c9">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">B</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> st <br/>( <font color="Olive" title="b1">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b1">B</font> &amp; <font color="Olive" title="b1">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">take </span>
<font color="Maroon" title="c10">B</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="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font> &amp; <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">thus </span><a NAME="E22:59_2_1_1_2"/>
<font color="Maroon" title="c10">B</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c8">p1</font>,<font color="Maroon" title="c9">p2</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E19:59_2_1_1_2"><span class="lab"><font color="Green" title="E61">A50</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"> ( <font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font> &amp; <font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><span class="kw">thus </span><a NAME="E23:59_2_1_1_2"/>
<font color="Maroon" title="c2">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c10">B</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E14:59_2_1_1_2"><span class="lab"><font color="Green" title="E60">A49</font></span></a>, <a class="txt" href="#E20:59_2_1_1_2"><span class="lab"><font color="Green" title="E62">A51</font></span></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></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="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font></span><br/><span class="kw">thus </span><a NAME="E24:59_2_1_1_2"/>
<font color="Maroon" title="c10">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font>
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E21:59_2_1_1_2"><span class="lab"><font color="Green" title="E63">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><span class="kw">end;</span></div></div><span class="kw">end;</span></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="59_2_2"><span class="kw">suppose </span></a><a NAME="E1:59_2_2"/>
 not <font color="Maroon" title="c2">C</font> is  <a href="zfmisc_1.html#V1" title="ZFMISC_1:attr.1">trivial</a> 
 ; <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">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )</span><br/><div class="add"><span class="kw">hence </span><a NAME="E2:59_2_2"/>
 ex <font color="Olive" title="b1">p1</font>, <font color="Olive" title="b2">p2</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> ex <font color="Olive" title="b3">B</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> st <br/>( <font color="Olive" title="b3">B</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">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b3">B</font> &amp; <font color="Olive" title="b3">B</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c1">D</font> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:59"><span class="lab"><font color="Green" title="E50">A1</font></span></a>, <a class="txt" href="#E5:59"><span class="lab"><font color="Green" title="E53">A31</font></span></a>, <a class="txt" href="#E3:59"><span class="lab"><font color="Green" title="E52">A3</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>
