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

<b>let </b><font color="Maroon" title="c10">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>

<b>thus </b><a NAME="E1:16"/>
( <font color="Maroon" title="c10">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a>  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> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> ) &amp; ( for <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>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> holds <br/> ex <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b3">P1</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="Olive" title="b4">P2</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="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">P2</font> &amp; <font color="Olive" title="b3">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> ) ) ) )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:16_1"/><i><font color="Green" title="E34">A1</font></i>: 
<font color="Maroon" title="c10">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</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="c10">P</font></span>)</span><b> such that </b><br/><a NAME="E3:16_1"/><i><font color="Green" title="E35">A2</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="ref" href="topreal2.html#D1" target="_self" title="TOPREAL2:def.1">Def1</a></i>;<br/>
<b>thus </b><a NAME="E4:16_1"/>
 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> st <br/>( <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E1:16_1"><i><font color="Green" title="E34">A1</font></i></a>, <a class="ref" href="topreal2.html#T4" target="_self" title="TOPREAL2:th.4">Th4</a></i>;<br/>

<b>let </b><font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c13">p2</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>



<b>assume </b><b>that </b><br/><a NAME="E5:16_1"/><i><font color="Green" title="E36">A3</font></i>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c13">p2</font>
 <b>and </b><br/><a NAME="E6:16_1"/><i><font color="Green" title="E37">A4</font></i>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 <b>and </b><br/><a NAME="E7:16_1"/><i><font color="Green" title="E38">A5</font></i>: 
<font color="Maroon" title="c13">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font>
 ;<br/>

<a NAME="E8:16_1"/><i><font color="Green" title="E39">A6</font></i>: 
<font color="Maroon" title="c11">f</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E9:16_1"/><i><font color="Green" title="E40">A7</font></i>: 
<font color="Maroon" title="c11">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E10:16_1"/><i><font color="Green" title="E41">A8</font></i>: 
(  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c10">P</font></span>)</span> &amp;  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16_1"><i><font color="Green" title="E35">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E11:16_1"/><b>then </b><i><font color="Green" title="E42">A9</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">f</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/>
<a NAME="E12:16_1"/><b>then </b><i><font color="Green" title="E43">A10</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/>
<a NAME="E13:16_1"/><i><font color="Green" title="E44">A11</font></i>: 
<font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="funct_1.html#T62" title="FUNCT_1:th.62">FUNCT_1:62</a></i>;<br/>
<a NAME="E14:16_1"/><i><font color="Green" title="E45">A12</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c11">f</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="ref" href="funct_1.html#T55" title="FUNCT_1:th.55">FUNCT_1:55</a></i>;<br/>
<a NAME="E15:16_1"/><i><font color="Green" title="E46">A13</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c10">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E16:16_1"/><i><font color="Green" title="E47">A14</font></i>: 
( <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> &amp; <font color="Maroon" title="c13">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E6:16_1"><i><font color="Green" title="E37">A4</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1"><i><font color="Green" title="E38">A5</font></i></a>, <a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>set </b><font color="Maroon" title="c14">q1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p1</font>;<br/>
<b>set </b><font color="Maroon" title="c15">q2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p2</font>;<br/>
<b>set </b><font color="Maroon" title="c16">RS</font> =  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> ;<br/>
<b>reconsider </b><font color="Maroon" title="c17">f</font> = <font color="Maroon" title="c11">f</font> as   <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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</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="c10">P</font></span>)</span> ;<br/>
<a NAME="E18:16_1"/><i><font color="Green" title="E48">A15</font></i>: 
(  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span> <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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a>, <a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">q1</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">p1</font>, <font color="Maroon" title="c19">q2</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c11">f</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">p2</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> ;<br/>
<a NAME="E20:16_1"/><i><font color="Green" title="E49">A16</font></i>: 
<font color="Maroon" title="c18">q1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c19">q2</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E5:16_1"><i><font color="Green" title="E36">A3</font></i></a>, <a class="txt" href="topreal2.html#E13:16_1"><i><font color="Green" title="E44">A11</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/>
<a NAME="E21:16_1"/><i><font color="Green" title="E50">A17</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E22:16_1"/><i><font color="Green" title="E51">A18</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E23:16_1"/><b>then </b>
( <font color="Maroon" title="c18">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  &amp; <font color="Maroon" title="c19">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  )
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c20">Q1</font>, <font color="Maroon" title="c21">Q2</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E25:16_1"/><i><font color="Green" title="E52">A19</font></i>: 
( <font color="Maroon" title="c20">Q1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font> &amp; <font color="Maroon" title="c21">Q2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font> )
 <b>and </b><br/><a NAME="E26:16_1"/><i><font color="Green" title="E53">A20</font></i>: 
(  <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c21">Q2</font> &amp; <font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 <b>by </b><i><a class="txt" href="topreal2.html#E20:16_1"><i><font color="Green" title="E49">A16</font></i></a>, <a class="ref" href="topreal2.html#T1" target="_self" title="TOPREAL2:th.1">Th1</a></i>;<br/>
<b>set </b><font color="Maroon" title="c22">P1</font> = <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">Q1</font>;<br/>
<b>set </b><font color="Maroon" title="c23">P2</font> = <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">Q2</font>;<br/>
<a NAME="E27:16_1"/><i><font color="Green" title="E54">A21</font></i>: 
( <font color="Maroon" title="c20">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font> &amp; <font color="Maroon" title="c21">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E28:16_1"/>
( <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font> &amp; <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font> &amp;  <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="c10">P</font></span>)</span> <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"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> )
 
<b>by </b><i><a class="ref" href="pre_topc.html#D9" title="PRE_TOPC:def.9">PRE_TOPC:def 9</a>, <a class="ref" href="relat_1.html#T144" title="RELAT_1:th.144">RELAT_1:144</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c24">P1</font> = <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c20">Q1</font>, <font color="Maroon" title="c25">P2</font> = <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <font color="Maroon" title="c21">Q2</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="topreal2.html#E27:16_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="relat_1.html#T152" title="RELAT_1:th.152">RELAT_1:152</a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/>
<b>take </b>
<font color="Maroon" title="c24">P1</font>
;<br/>

<b>take </b>
<font color="Maroon" title="c25">P2</font>
;<br/>

<i><font color="Green" title="E55">A22</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c20">Q1</font></span>)</span> = 
<font color="Maroon" title="c24">P1</font>
<b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>
<br/>.= 
 <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="c24">P1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
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="c24">P1</font></span>)</span>

;<br/>
<i><font color="Green" title="E56">A23</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c20">Q1</font></span>)</span> = 
<a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font>
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c20">Q1</font>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
<br/>.= 
 <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="c20">Q1</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c26">F1</font> = <font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c20">Q1</font> as   <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="c20">Q1</font></span>)</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="c24">P1</font></span>)</span> <b>by </b><i><a class="txt" href="topreal2.html#E30:16_1"><i><font color="Green" title="E55">A22</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E33:16_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"><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="c20">Q1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E34:16_1"/><b>then </b>
 <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="c20">Q1</font></span>)</span> <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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E35:16_1"/><b>then </b><i><font color="Green" title="E57">A24</font></i>: 
<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="c20">Q1</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/>
<a NAME="E36:16_1"/><i><font color="Green" title="E58">A25</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c26">F1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c24">P1</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E30:16_1"><i><font color="Green" title="E55">A22</font></i></a></i>;<br/>
<a NAME="E37:16_1"/><i><font color="Green" title="E59">A26</font></i>: 
<font color="Maroon" title="c26">F1</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<a NAME="E38:16_1"/>
( <font color="Maroon" title="c20">Q1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">P1</font> &amp; <font color="Maroon" title="c21">Q2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c25">P2</font> &amp; <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Q1</font> &amp; <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c25">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">Q2</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E27:16_1"><i><font color="Green" title="E54">A21</font></i></a>, <a class="ref" href="funct_1.html#T146" title="FUNCT_1:th.146">FUNCT_1:146</a>, <a class="ref" href="funct_1.html#T152" title="FUNCT_1:th.152">FUNCT_1:152</a></i>;<br/>
<a NAME="E39:16_1"/><b>then </b><i><font color="Green" title="E60">A27</font></i>: 
( <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c24">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c20">Q1</font> &amp; <font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c25">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Q2</font> )
 
<b>by </b><i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<a NAME="E40:16_1"/>
for <font color="Olive" title="b1">R</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"><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="c24">P1</font></span>)</span>  st <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Maroon" title="c26">F1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c27">R</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"><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="c24">P1</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_3"/>
<font color="Maroon" title="c27">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c28">S1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:16_1_3"/><i><font color="Green" title="E61">A28</font></i>: 
<font color="Maroon" title="c28">S1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:16_1_3"/><i><font color="Green" title="E62">A29</font></i>: 
<font color="Maroon" title="c27">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c28">S1</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="c24">P1</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E5:16_1_3"/>
<font color="Maroon" title="c28">S1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> is   <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="c10">P</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c29">S2</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c28">S1</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="c10">P</font></span>)</span> ;<br/>
<i><font color="Green" title="E63">A30</font></i>: <font color="Maroon" title="c26">F1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">R</font> = 
<font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">R</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T139" title="FUNCT_1:th.139">FUNCT_1:139</a></i>
<br/>.= 
<font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c28">S1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p4">(<span class="default"><span class="p5">(<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="c24">P1</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E4:16_1_3"><i><font color="Green" title="E62">A29</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c28">S1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font>
<b>by </b><i><a class="txt" href="topreal2.html#E39:16_1"><i><font color="Green" title="E60">A27</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c28">S1</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="c20">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c28">S1</font></span>)</span> <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="c20">Q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c28">S1</font></span>)</span></span>)</span> <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="c20">Q1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
;<br/>
<a NAME="E8:16_1_3"/>
<font color="Maroon" title="c29">S2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="topreal2.html#E3:16_1_3"><i><font color="Green" title="E61">A28</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E9:16_1_3"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c29">S2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E9:16_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_3"/>
<font color="Maroon" title="c26">F1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c27">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="topreal2.html#E35:16_1"><i><font color="Green" title="E57">A24</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1_3"><i><font color="Green" title="E63">A30</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E41:16_1"/><b>then </b><i><font color="Green" title="E61">A31</font></i>: 
<font color="Maroon" title="c26">F1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c27">ff</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c20">Q1</font></span>)</span><b> such that </b><br/><a NAME="E43:16_1"/><i><font color="Green" title="E62">A32</font></i>: 
<font color="Maroon" title="c27">ff</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E44:16_1"/>
( <font color="Maroon" title="c27">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q1</font> &amp; <font color="Maroon" title="c27">ff</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="c19">q2</font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E45:16_1"/><i><font color="Green" title="E63">A33</font></i>: 
 <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="ref" href="heine.html#T11" title="HEINE:th.11">HEINE:11</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c28">Q1'</font> = <font color="Maroon" title="c20">Q1</font>, <font color="Maroon" title="c29">Q2'</font> = <font color="Maroon" title="c21">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> ;<br/>
<a NAME="E47:16_1"/>
( <font color="Maroon" title="c27">ff</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c27">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c20">Q1</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E43:16_1"><i><font color="Green" title="E62">A32</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E48:16_1"/><b>then </b><i><font color="Green" title="E64">A34</font></i>: 
<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="c28">Q1'</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E45:16_1"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<a NAME="E49:16_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="c24">P1</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<a NAME="E50:16_1"/><b>then </b><i><font color="Green" title="E65">A35</font></i>: 
<font color="Maroon" title="c26">F1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E31:16_1"><i><font color="Green" title="E56">A23</font></i></a>, <a class="txt" href="topreal2.html#E36:16_1"><i><font color="Green" title="E58">A25</font></i></a>, <a class="txt" href="topreal2.html#E37:16_1"><i><font color="Green" title="E59">A26</font></i></a>, <a class="txt" href="topreal2.html#E41:16_1"><i><font color="Green" title="E61">A31</font></i></a>, <a class="txt" href="topreal2.html#E48:16_1"><i><font color="Green" title="E64">A34</font></i></a>, <a class="ref" href="compts_1.html#T26" title="COMPTS_1:th.26">COMPTS_1:26</a></i>;<br/>
<a NAME="E51:16_1"/>
( <font color="Maroon" title="c18">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Q1</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E52:16_1"/><b>then </b><i><font color="Green" title="E66">A36</font></i>: 
<font color="Maroon" title="c18">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<i><font color="Green" title="E67">A37</font></i>: <font color="Maroon" title="c12">p1</font> = 
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q1</font>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c26">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q1</font>
<b>by </b><i><a class="txt" href="topreal2.html#E52:16_1"><i><font color="Green" title="E66">A36</font></i></a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>
;<br/>
<a NAME="E54:16_1"/>
( <font color="Maroon" title="c19">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c20">Q1</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E55:16_1"/><b>then </b><i><font color="Green" title="E68">A38</font></i>: 
<font color="Maroon" title="c19">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c20">Q1</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<font color="Maroon" title="c13">p2</font> = 
<font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">q2</font>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>
<br/>.= 
<font color="Maroon" title="c26">F1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">q2</font>
<b>by </b><i><a class="txt" href="topreal2.html#E55:16_1"><i><font color="Green" title="E68">A38</font></i></a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>
;<br/>
<b>hence </b><a NAME="E57:16_1"/>
<font color="Maroon" title="c24">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font>
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="topreal2.html#E50:16_1"><i><font color="Green" title="E65">A35</font></i></a>, <a class="txt" href="topreal2.html#E53:16_1"><i><font color="Green" title="E67">A37</font></i></a>, <a class="ref" href="topreal2.html#T3" target="_self" title="TOPREAL2:th.3">Th3</a></i>;<br/>

<i><font color="Green" title="E69">A39</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">Q2</font></span>)</span> = 
<font color="Maroon" title="c25">P2</font>
<b>by </b><i><a class="ref" href="relat_1.html#T148" title="RELAT_1:th.148">RELAT_1:148</a></i>
<br/>.= 
 <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="c25">P2</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
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="c25">P2</font></span>)</span>

;<br/>
<i><font color="Green" title="E70">A40</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">Q2</font></span>)</span> = 
<a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a>  <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font>
<b>by </b><i><a class="txt" href="topreal2.html#E21:16_1"><i><font color="Green" title="E50">A17</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="ref" href="relat_1.html#T90" title="RELAT_1:th.90">RELAT_1:90</a></i>
<br/>.= 
<font color="Maroon" title="c21">Q2</font>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T21" title="XBOOLE_1:th.21">XBOOLE_1:21</a></i>
<br/>.= 
 <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="c21">Q2</font></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
;<br/>
<b>then reconsider </b><font color="Maroon" title="c30">F2</font> = <font color="Maroon" title="c17">f</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c21">Q2</font> as   <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="c21">Q2</font></span>)</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="c25">P2</font></span>)</span> <b>by </b><i><a class="txt" href="topreal2.html#E58:16_1"><i><font color="Green" title="E69">A39</font></i></a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11" title="RELSET_1:th.11">RELSET_1:11</a></i>;<br/>
<a NAME="E61:16_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"><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="c21">Q2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c21">Q2</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E62:16_1"/><b>then </b>
 <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="c21">Q2</font></span>)</span> <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> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> </span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="xboole_1.html#T7" title="XBOOLE_1:th.7">XBOOLE_1:7</a></i>;<br/>
<a NAME="E63:16_1"/><b>then </b><i><font color="Green" title="E71">A41</font></i>: 
<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="c21">Q2</font> is    <a href="pre_topc.html#M1" title="PRE_TOPC:mode.1">SubSpace</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="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <a href="topreal1.html#K2" title="TOPREAL1:func.2">R^2-unit_square</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T4" title="TOPMETR:th.4">TOPMETR:4</a></i>;<br/>
<a NAME="E64:16_1"/><i><font color="Green" title="E72">A42</font></i>: 
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c30">F2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c25">P2</font></span>)</span>
 
<b>by </b><i><a class="txt" href="topreal2.html#E58:16_1"><i><font color="Green" title="E69">A39</font></i></a></i>;<br/>
<a NAME="E65:16_1"/><i><font color="Green" title="E73">A43</font></i>: 
<font color="Maroon" title="c30">F2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_1.html#T84" title="FUNCT_1:th.84">FUNCT_1:84</a></i>;<br/>
<a NAME="E66:16_1"/>
for <font color="Olive" title="b1">R</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"><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="c25">P2</font></span>)</span>  st <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Maroon" title="c30">F2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="16_1_8"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c31">R</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"><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="c25">P2</font></span>)</span>;<br/>

<b>assume </b><a NAME="E1:16_1_8"/>
<font color="Maroon" title="c31">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c32">S1</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:16_1_8"/><i><font color="Green" title="E74">A44</font></i>: 
<font color="Maroon" title="c32">S1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>and </b><br/><a NAME="E4:16_1_8"/><i><font color="Green" title="E75">A45</font></i>: 
<font color="Maroon" title="c31">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c32">S1</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="c25">P2</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E5:16_1_8"/>
<font color="Maroon" title="c32">S1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> is   <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="c10">P</font></span>)</span>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c33">S2</font> = <span class="p1">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c32">S1</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="c10">P</font></span>)</span> ;<br/>
<i><font color="Green" title="E76">A46</font></i>: <font color="Maroon" title="c30">F2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c31">R</font> = 
<font color="Maroon" title="c21">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c31">R</font></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T139" title="FUNCT_1:th.139">FUNCT_1:139</a></i>
<br/>.= 
<font color="Maroon" title="c21">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c32">S1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p3">(<span class="default"><a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p4">(<span class="default"><span class="p5">(<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="c25">P2</font></span>)</span></span>)</span></span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E4:16_1_8"><i><font color="Green" title="E75">A45</font></i></a>, <a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c32">S1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font>
<b>by </b><i><a class="txt" href="topreal2.html#E39:16_1"><i><font color="Green" title="E60">A27</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c32">S1</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="c21">Q2</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font></span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T16" title="XBOOLE_1:th.16">XBOOLE_1:16</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c32">S1</font></span>)</span> <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="c21">Q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c32">S1</font></span>)</span></span>)</span> <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="c21">Q2</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T168" title="RELAT_1:th.168">RELAT_1:168</a></i>
;<br/>
<a NAME="E8:16_1_8"/>
<font color="Maroon" title="c33">S2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="topreal2.html#E3:16_1_8"><i><font color="Green" title="E74">A44</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>
<a NAME="E9:16_1_8"/><b>then </b>
<font color="Maroon" title="c17">f</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c33">S2</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E9:16_1"><i><font color="Green" title="E40">A7</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<b>hence </b><a NAME="E10:16_1_8"/>
<font color="Maroon" title="c30">F2</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c31">R</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 <b>by </b><i><a class="txt" href="topreal2.html#E63:16_1"><i><font color="Green" title="E71">A41</font></i></a>, <a class="txt" href="topreal2.html#E7:16_1_8"><i><font color="Green" title="E76">A46</font></i></a>, <a class="ref" href="pre_topc.html#T43" title="PRE_TOPC:th.43">PRE_TOPC:43</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E67:16_1"/><b>then </b><i><font color="Green" title="E74">A47</font></i>: 
<font color="Maroon" title="c30">F2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c31">ff</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c21">Q2</font></span>)</span><b> such that </b><br/><a NAME="E69:16_1"/><i><font color="Green" title="E75">A48</font></i>: 
<font color="Maroon" title="c31">ff</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E70:16_1"/>
( <font color="Maroon" title="c31">ff</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">q1</font> &amp; <font color="Maroon" title="c31">ff</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="c19">q2</font> )
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E71:16_1"/>
( <font color="Maroon" title="c31">ff</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c31">ff</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</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="c21">Q2</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E69:16_1"><i><font color="Green" title="E75">A48</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E72:16_1"/><b>then </b><i><font color="Green" title="E76">A49</font></i>: 
<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="c29">Q2'</font> is <a href="compts_1.html#V2" title="COMPTS_1:attr.2">compact</a>
 
<b>by </b><i><a class="txt" href="topreal2.html#E45:16_1"><i><font color="Green" title="E63">A33</font></i></a>, <a class="ref" href="compts_1.html#T23" title="COMPTS_1:th.23">COMPTS_1:23</a></i>;<br/>
<a NAME="E73:16_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="c25">P2</font> <a href="compts_1.html#NR2" title="COMPTS_1:NR.2">is_T2</a> 
 
<b>by </b><i><a class="ref" href="topmetr.html#T3" title="TOPMETR:th.3">TOPMETR:3</a></i>;<br/>
<a NAME="E74:16_1"/><b>then </b><i><font color="Green" title="E77">A50</font></i>: 
<font color="Maroon" title="c30">F2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="txt" href="topreal2.html#E59:16_1"><i><font color="Green" title="E70">A40</font></i></a>, <a class="txt" href="topreal2.html#E64:16_1"><i><font color="Green" title="E72">A42</font></i></a>, <a class="txt" href="topreal2.html#E65:16_1"><i><font color="Green" title="E73">A43</font></i></a>, <a class="txt" href="topreal2.html#E67:16_1"><i><font color="Green" title="E74">A47</font></i></a>, <a class="txt" href="topreal2.html#E72:16_1"><i><font color="Green" title="E76">A49</font></i></a>, <a class="ref" href="compts_1.html#T26" title="COMPTS_1:th.26">COMPTS_1:26</a></i>;<br/>
<a NAME="E75:16_1"/>
( <font color="Maroon" title="c18">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">Q2</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E76:16_1"/><b>then </b><i><font color="Green" title="E78">A51</font></i>: 
<font color="Maroon" title="c18">q1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E77:16_1"/><i><font color="Green" title="E79">A52</font></i>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q1</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E78:16_1"/><b>then </b><i><font color="Green" title="E80">A53</font></i>: 
<font color="Maroon" title="c12">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c18">q1</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E76:16_1"><i><font color="Green" title="E78">A51</font></i></a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>;<br/>
<a NAME="E79:16_1"/>
( <font color="Maroon" title="c19">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> &amp; <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c18">q1</font>,<font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c21">Q2</font> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="tarski.html#D2" title="TARSKI:def.2">TARSKI:def 2</a>, <a class="ref" href="xboole_1.html#T17" title="XBOOLE_1:th.17">XBOOLE_1:17</a></i>;<br/>
<a NAME="E80:16_1"/><b>then </b><i><font color="Green" title="E81">A54</font></i>: 
<font color="Maroon" title="c19">q2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c17">f</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="ref" href="xboole_0.html#D3" title="XBOOLE_0:def.3">XBOOLE_0:def 3</a></i>;<br/>
<a NAME="E81:16_1"/><i><font color="Green" title="E82">A55</font></i>: 
<font color="Maroon" title="c13">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c17">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">q2</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="txt" href="topreal2.html#E11:16_1"><i><font color="Green" title="E42">A9</font></i></a>, <a class="txt" href="topreal2.html#E12:16_1"><i><font color="Green" title="E43">A10</font></i></a>, <a class="txt" href="topreal2.html#E16:16_1"><i><font color="Green" title="E47">A14</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">FUNCT_1:57</a></i>;<br/>
<a NAME="E82:16_1"/><b>then </b>
<font color="Maroon" title="c13">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c30">F2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c19">q2</font>
 
<b>by </b><i><a class="txt" href="topreal2.html#E80:16_1"><i><font color="Green" title="E81">A54</font></i></a>, <a class="ref" href="funct_1.html#T71" title="FUNCT_1:th.71">FUNCT_1:71</a></i>;<br/>
<b>hence </b><a NAME="E83:16_1"/>
<font color="Maroon" title="c25">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font>
 <b>by </b><i><a class="txt" href="topreal2.html#E25:16_1"><i><font color="Green" title="E52">A19</font></i></a>, <a class="txt" href="topreal2.html#E74:16_1"><i><font color="Green" title="E77">A50</font></i></a>, <a class="txt" href="topreal2.html#E78:16_1"><i><font color="Green" title="E80">A53</font></i></a>, <a class="ref" href="topreal2.html#T3" target="_self" title="TOPREAL2:th.3">Th3</a></i>;<br/>

<b>thus </b><font color="Maroon" title="c10">P</font> = 
<font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c21">Q2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E10:16_1"><i><font color="Green" title="E41">A8</font></i></a>, <a class="txt" href="topreal2.html#E15:16_1"><i><font color="Green" title="E46">A13</font></i></a>, <a class="txt" href="topreal2.html#E22:16_1"><i><font color="Green" title="E51">A18</font></i></a>, <a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="relat_1.html#T146" title="RELAT_1:th.146">RELAT_1:146</a></i>
<br/>.= 
<font color="Maroon" title="c24">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c25">P2</font>
<b>by </b><i><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
;<br/>

<b>thus </b><font color="Maroon" title="c24">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c25">P2</font> = 
<font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><font color="Maroon" title="c20">Q1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c21">Q2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E8:16_1"><i><font color="Green" title="E39">A6</font></i></a>, <a class="ref" href="funct_1.html#T121" title="FUNCT_1:th.121">FUNCT_1:121</a></i>
<br/>.= 
<font color="Maroon" title="c17">f</font> <a href="relset_1.html#K10" title="RELSET_1:func.10">.:</a> <span class="p1">(<span class="default"><span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c18">q1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p2"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c19">q2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E26:16_1"><i><font color="Green" title="E53">A20</font></i></a>, <a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c18">q1</font></span>)</span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c19">q2</font></span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T153" title="RELAT_1:th.153">RELAT_1:153</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1">(<span class="default"><a href="relat_1.html#K11" title="RELAT_1:func.11">Im</a> <font color="Maroon" title="c17">f</font>,<font color="Maroon" title="c19">q2</font></span>)</span>
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="topreal2.html#E77:16_1"><i><font color="Green" title="E79">A52</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c12">p1</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> <a href="xboole_0.html#K2" title="XBOOLE_0:func.2">\/</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
<b>by </b><i><a class="txt" href="topreal2.html#E14:16_1"><i><font color="Green" title="E45">A12</font></i></a>, <a class="txt" href="topreal2.html#E18:16_1"><i><font color="Green" title="E48">A15</font></i></a>, <a class="txt" href="topreal2.html#E81:16_1"><i><font color="Green" title="E82">A55</font></i></a>, <a class="ref" href="funct_1.html#T117" title="FUNCT_1:th.117">FUNCT_1:117</a></i>
<br/>.= 
<span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c12">p1</font>,<font color="Maroon" title="c13">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span>
<b>by </b><i><a class="ref" href="enumset1.html#T41" title="ENUMSET1:th.41">ENUMSET1:41</a></i>
;<br/>


</div><b>end;</b></div>

<b>given </b><font color="Maroon" title="c11">p1</font>, <font color="Maroon" title="c12">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><b> such that </b><a NAME="E3:16"/><i><font color="Green" title="E34">A56</font></i>: 
( <font color="Maroon" title="c11">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c12">p2</font> &amp; <font color="Maroon" title="c11">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Maroon" title="c12">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> )
 ;<br/>

<b>assume </b><a NAME="E4:16"/>
for <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>  st <font color="Olive" title="b1">p1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Olive" title="b2">p2</font> &amp; <font color="Olive" title="b1">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> &amp; <font color="Olive" title="b2">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P</font> holds <br/> ex <font color="Olive" title="b3">P1</font>, <font color="Olive" title="b4">P2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b3">P1</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="Olive" title="b4">P2</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="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b4">P2</font> &amp; <font color="Olive" title="b3">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b4">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Olive" title="b1">p1</font>,<font color="Olive" title="b2">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 ;<br/>

<a NAME="E5:16"/><b>then </b>
 ex <font color="Olive" title="b1">P1</font>, <font color="Olive" title="b2">P2</font> being non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> st <br/>( <font color="Olive" title="b1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp; <font color="Olive" title="b2">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font> &amp; <font color="Maroon" title="c10">P</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b1">P1</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Olive" title="b2">P2</font> &amp; <font color="Olive" title="b1">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Olive" title="b2">P2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="tarski.html#K2" title="TARSKI:func.2">{</a><span class="default"><font color="Maroon" title="c11">p1</font>,<font color="Maroon" title="c12">p2</font></span><a href="tarski.html#K2" title="TARSKI:func.2">}</a></span> )
 
<b>by </b><i><a class="txt" href="topreal2.html#E3:16"><i><font color="Green" title="E34">A56</font></i></a></i>;<br/>
<b>hence </b><a NAME="E6:16"/>
<font color="Maroon" title="c10">P</font> <a href="topreal2.html#NR1" title="TOPREAL2:NR.1">is_simple_closed_curve</a> 
 <b>by </b><i><a class="txt" href="topreal2.html#E3:16"><i><font color="Green" title="E34">A56</font></i></a>, <a class="txt" href="topreal2.html#E32"><i><font color="Green" title="E32">Lm27</font></i></a></i>;<br/>


</div>
