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

<b>let </b><font color="Maroon" title="c7">p1</font>, <font color="Maroon" title="c8">p2</font>, <font color="Maroon" title="c9">p</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>let </b><font color="Maroon" title="c10">A</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c11">r</font> be  non <a href="xxreal_0.html#V3" title="XXREAL_0:attr.3">negative</a> <a href="xreal_0.html#V1" title="XREAL_0:attr.1">real</a>  <a href="ordinal1.html#NM1" title="ORDINAL1:NM.1">number</a> ;<br/>









<b>set </b><font color="Maroon" title="c12">D</font> =  <a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font>;<br/>
<b>assume </b><b>that </b><br/><a NAME="E1:104"/><i><font color="Green" title="E79">A1</font></i>: 
<font color="Maroon" title="c10">A</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c7">p1</font>,<font color="Maroon" title="c8">p2</font>
 <b>and </b><br/><a NAME="E2:104"/><i><font color="Green" title="E80">A2</font></i>: 
<font color="Maroon" title="c10">A</font> is   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></span>)</span>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c13">A1</font> = <font color="Maroon" title="c10">A</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="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></span>)</span> <b>by </b><i><a class="txt" href="jordan.html#E1:104"><i><font color="Green" title="E79">A1</font></i></a>, <a class="txt" href="jordan.html#E2:104"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">A2</font> = <font color="Maroon" title="c10">A</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="jordan.html#E1:104"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>set </b><font color="Maroon" title="c15">TA</font> = <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="c14">A2</font>;<br/>
<b>consider </b><font color="Maroon" title="c16">h</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="c14">A2</font></span>)</span><b> such that </b><br/><a NAME="E6:104"/><i><font color="Green" title="E81">A3</font></i>: 
<font color="Maroon" title="c16">h</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>and </b><br/><a NAME="E7:104"/>
<font color="Maroon" title="c16">h</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">p1</font>
 <b>and </b><br/><a NAME="E8:104"/>
<font color="Maroon" title="c16">h</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="c8">p2</font>
 <b>by </b><i><a class="txt" href="jordan.html#E1:104"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E9:104"/><i><font color="Green" title="E82">A4</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 
<b>by </b><i><a class="ref" href="treal_1.html#T20" title="TREAL_1:th.20">TREAL_1:20</a></i>;<br/>
<a NAME="E10:104"/><b>then </b><i><font color="Green" title="E83">A5</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c17">hh</font> = <font color="Maroon" title="c16">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</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="c14">A2</font></span>)</span> <b>by </b><i><a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<a NAME="E12:104"/><i><font color="Green" title="E84">A6</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="c14">A2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">A1</font>
 
<b>by </b><i><a class="ref" href="topalg_5.html#T4" title="TOPALG_5:th.4">TOPALG_5:4</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c18">f</font> = <span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> 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="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c13">A1</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span> ;<br/>
<a NAME="E14:104"/>
<font color="Maroon" title="c10">A</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E1:104"><i><font color="Green" title="E79">A1</font></i></a>, <a class="ref" href="jordan6.html#T12" title="JORDAN6:th.12">JORDAN6:12</a></i>;<br/>
<a NAME="E15:104"/><b>then </b><i><font color="Green" title="E85">A7</font></i>: 
<font color="Maroon" title="c13">A1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="tsep_1.html#T12" title="TSEP_1:th.12">TSEP_1:12</a></i>;<br/>
<a NAME="E16:104"/>
<font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E6:104"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E17:104"/><b>then </b>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E10:104"><i><font color="Green" title="E83">A5</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c19">g</font> being  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></span>)</span>,<span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span><b> such that </b><br/><a NAME="E19:104"/><i><font color="Green" title="E86">A8</font></i>: 
<font color="Maroon" title="c19">g</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c13">A1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c18">f</font>
 <b>by </b><i><a class="txt" href="jordan.html#E12:104"><i><font color="Green" title="E84">A6</font></i></a>, <a class="txt" href="jordan.html#E15:104"><i><font color="Green" title="E85">A7</font></i></a>, <a class="ref" href="tietze.html#T25" title="TIETZE:th.25">TIETZE:25</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c20">R</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c19">g</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</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="c10">A</font></span>)</span> ;<br/>
<b>take </b>
<font color="Maroon" title="c20">R</font>
;<br/>

<a NAME="E21:104"/><i><font color="Green" title="E87">A9</font></i>: 
<font color="Maroon" title="c17">hh</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E6:104"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E22:104"/>
<span class="p1">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p2">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E9:104"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E23:104"/><b>then </b>
<font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan.html#E21:104"><i><font color="Green" title="E87">A9</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<b>hence </b><a NAME="E24:104"/>
<font color="Maroon" title="c20">R</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>

<a NAME="E25:104"/><i><font color="Green" title="E88">A10</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.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="c14">A2</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c13">A1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/>
<a NAME="E26:104"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c20">R</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"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></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="E27:104"/><b>then </b><i><font color="Green" title="E89">A11</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="c20">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="txt" href="jordan.html#E2:104"><i><font color="Green" title="E80">A2</font></i></a>, <a class="ref" href="relat_1.html#T91" title="RELAT_1:th.91">RELAT_1:91</a></i>;<br/>
<a NAME="E28:104"/><i><font color="Green" title="E90">A12</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">A</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T71" title="RELAT_1:th.71">RELAT_1:71</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="104_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c21">a</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:104_1"/><i><font color="Green" title="E91">A13</font></i>: 
<font color="Maroon" title="c21">a</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="c20">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">A</font></span>)</span>
 ;<br/><a NAME="E2:104_1"/><i><font color="Green" title="E92">A14</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c19">g</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"><a href="brouwer.html#K2" title="BROUWER:func.2">Tdisk</a> <font color="Maroon" title="c9">p</font>,<font color="Maroon" title="c11">r</font></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="E3:104_1"/><i><font color="Green" title="E93">A15</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c14">A2</font></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="E4:104_1"/><i><font color="Green" title="E94">A16</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan.html#E9:104"><i><font color="Green" title="E82">A4</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="E5:104_1"/><i><font color="Green" title="E95">A17</font></i>: 
 <a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p1">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a>
 <b>by </b><i><a class="txt" href="jordan.html#E9:104"><i><font color="Green" title="E82">A4</font></i></a>, <a class="ref" href="jordan.html#T27" target="_self" title="JORDAN:th.27">Th27</a></i>;<br/><a NAME="E6:104_1"/><i><font color="Green" title="E96">A18</font></i>: 
( <font color="Maroon" title="c17">hh</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a> &amp; <font color="Maroon" title="c17">hh</font> is <a href="funct_2.html#V2" title="FUNCT_2:attr.2">onto</a> )
 <b>by </b><i><a class="txt" href="jordan.html#E6:104"><i><font color="Green" title="E81">A3</font></i></a>, <a class="ref" href="jordan.html#T27" target="_self" title="JORDAN:th.27">Th27</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><i><font color="Green" title="E97">A19</font></i>: <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p3">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p5">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="relat_1.html#T55" title="RELAT_1:th.55">RELAT_1:55</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><a href="topgrp_1.html#K4" title="TOPGRP_1:func.4">id</a> <span class="p3">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span>)</span></span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E4:104_1"><i><font color="Green" title="E94">A16</font></i></a>, <a class="txt" href="jordan.html#E5:104_1"><i><font color="Green" title="E95">A17</font></i></a>, <a class="ref" href="grcat_1.html#T56" title="GRCAT_1:th.56">GRCAT_1:56</a></i>
<br/>.= 
<font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#T23" title="FUNCT_2:th.23">FUNCT_2:23</a></i>
<br/>.= 
 <a href="topgrp_1.html#K4" title="TOPGRP_1:func.4">id</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="c14">A2</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E6:104_1"><i><font color="Green" title="E96">A18</font></i></a>, <a class="ref" href="grcat_1.html#T56" title="GRCAT_1:th.56">GRCAT_1:56</a></i>
;<br/><b>thus </b><span class="p1">(<span class="default"><font color="Maroon" title="c20">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font> = 
<font color="Maroon" title="c20">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font>
<b>by </b><i><a class="txt" href="jordan.html#E27:104"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="jordan.html#E1:104_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E1:104_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="jordan.html#E2:104_1"><i><font color="Green" title="E92">A14</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="treal_1.html#K3" title="TREAL_1:func.3">L[01]</a> <span class="p4">(<span class="default"><a href="treal_1.html#K1" title="TREAL_1:func.1">(#)</a> <span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1</span>)</span>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="real_1.html#K1" title="REAL_1:func.1">-</a> 1</span>)</span>,1 <a href="treal_1.html#K2" title="TREAL_1:func.2">(#)</a> </span>)</span></span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <span class="p3">(<span class="default"><font color="Maroon" title="c17">hh</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan.html#E19:104"><i><font color="Green" title="E86">A8</font></i></a>, <a class="txt" href="jordan.html#E27:104"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="jordan.html#E1:104_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="ref" href="funct_1.html#T72" title="FUNCT_1:th.72">FUNCT_1:72</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c10">A</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c21">a</font>
<b>by </b><i><a class="txt" href="jordan.html#E25:104"><i><font color="Green" title="E88">A10</font></i></a>, <a class="txt" href="jordan.html#E27:104"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="jordan.html#E1:104_1"><i><font color="Green" title="E91">A13</font></i></a>, <a class="txt" href="jordan.html#E3:104_1"><i><font color="Green" title="E93">A15</font></i></a>, <a class="txt" href="jordan.html#E7:104_1"><i><font color="Green" title="E97">A19</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:104"/>
<font color="Maroon" title="c20">R</font> <a href="partfun1.html#K2" title="PARTFUN1:func.2">|</a> <font color="Maroon" title="c10">A</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="partfun1.html#K6" title="PARTFUN1:func.6">id</a> <font color="Maroon" title="c10">A</font>
 <b>by </b><i><a class="txt" href="jordan.html#E27:104"><i><font color="Green" title="E89">A11</font></i></a>, <a class="txt" href="jordan.html#E28:104"><i><font color="Green" title="E90">A12</font></i></a>, <a class="ref" href="funct_1.html#T9" title="FUNCT_1:th.9">FUNCT_1:9</a></i>;<br/>


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