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

<b>let </b><font color="Maroon" title="c1">P1</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="c2">r</font> be  <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>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">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><a NAME="E1:56"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  &amp; <font color="Maroon" title="c1">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c5">P1'</font> = <font color="Maroon" title="c1">P1</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="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">TOPREAL1:4</a></i>;<br/>
<b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">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="c5">P1'</font></span>)</span><b> such that </b><br/><a NAME="E4:56"/><i><font color="Green" title="E38">A2</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:56"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E5:56"/><i><font color="Green" title="E39">A3</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P1</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/>
<b>then reconsider </b><font color="Maroon" title="c7">f1</font> = <font color="Maroon" title="c6">f</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9" title="FUNCT_2:th.9">FUNCT_2:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c8">f2</font> = <font color="Maroon" title="c7">f1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon" title="c9">proj11</font> =  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>, <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c10">proj12</font> =  <a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</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="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">g1</font> = <font color="Maroon" title="c9">proj11</font> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c7">f1</font> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> , <a href="numbers.html#K1" title="NUMBERS:func.1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">g</font> = <font color="Maroon" title="c11">g1</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> ,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E12:56"/>
<font color="Maroon" title="c6">f</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E4:56"><i><font color="Green" title="E38">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="E13:56"/><b>then </b><i><font color="Green" title="E40">A4</font></i>: 
<font color="Maroon" title="c8">f2</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self" title="JORDAN6:th.4">Th4</a></i>;<br/>
<a NAME="E14:56"/>
<font color="Maroon" title="c10">proj12</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal5.html#T16" title="TOPREAL5:th.16">TOPREAL5:16</a></i>;<br/>
<a NAME="E15:56"/><b>then </b><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c12">g</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E13:56"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tops_2.html#T58" title="TOPS_2:th.58">TOPS_2:58</a></i>;<br/>
<a NAME="E16:56"/><i><font color="Green" title="E42">A6</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E17:56"/><b>then </b><i><font color="Green" title="E43">A7</font></i>: 
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E18:56"/><i><font color="Green" title="E44">A8</font></i>: 
1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan6.html#E16:56"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<i><font color="Green" title="E45">A9</font></i>: <font color="Maroon" title="c12">g</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="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c3">p1</font>
<b>by </b><i><a class="txt" href="jordan6.html#E4:56"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan6.html#E17:56"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c3">p1</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
;<br/>
<i><font color="Green" title="E46">A10</font></i>: <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 = 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c4">p2</font>
<b>by </b><i><a class="txt" href="jordan6.html#E4:56"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan6.html#E18:56"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon" title="c4">p2</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon" title="c13">P1'</font> = <font color="Maroon" title="c5">P1'</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="E22:56"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c13">P1'</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="txt" href="jordan6.html#E1:56"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="jordan6.html#T12" target="_self" title="JORDAN6:th.12">Th12</a></i>;<br/>
<a NAME="E23:56"/><i><font color="Green" title="E48">A12</font></i>: 
 <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self" title="JORDAN6:th.7">Th7</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="56_3"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:56_3_1"/>
( <font color="Maroon" title="c12">g</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="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 or <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> or <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K22" title="BORSUK_1:func.22">1[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> or ( <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K22" title="BORSUK_1:func.22">1[01]</a>  ) )
 </span><b>by </b><i><a class="txt" href="jordan6.html#E1:56"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E19:56"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="jordan6.html#E20:56"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_1"><b>case </b></a><a NAME="E1:56_3_1_1"/>
<font color="Maroon" title="c12">g</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="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:56_3_1_1"/><b>then </b><i><font color="Green" title="E49">A13</font></i>: 
<font color="Maroon" title="c12">g</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="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:56"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E19:56"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="jordan6.html#E20:56"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="xxreal_0.html#T1" title="XXREAL_0:th.1">XXREAL_0:1</a></i>;<br/><a NAME="E3:56_3_1_1"/><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:56"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:56_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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/><font color="Maroon" title="c14">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font>
<b>by </b><i><a class="txt" href="jordan6.html#E17:56"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E2:56_3_1_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><a NAME="E7:56_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E8:56_3_1_1"/>
<font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E3:56_3_1_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_2"><b>case </b></a><a NAME="E1:56_3_1_2"/><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 ;<br/><div class="add"><a NAME="E2:56_3_1_2"/><i><font color="Green" title="E50">A16</font></i>: 
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:56"><i><font color="Green" title="E43">A7</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:56_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  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/><font color="Maroon" title="c14">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> </span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font>
<b>by </b><i><a class="txt" href="jordan6.html#E17:56"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E1:56_3_1_2"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="borsuk_1.html#D17" title="BORSUK_1:def.17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><a NAME="E6:56_3_1_2"/><b>then </b>
<font color="Maroon" title="c6">f</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#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:56_3_1_2"/>
<font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E2:56_3_1_2"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_3"><b>case </b></a><a NAME="E1:56_3_1_3"/><i><font color="Green" title="E49">A17</font></i>: 
<font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K22" title="BORSUK_1:func.22">1[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font>
 ;<br/><div class="add"><a NAME="E2:56_3_1_3"/><i><font color="Green" title="E50">A18</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E3:56_3_1_3"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c14">p</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 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/><font color="Maroon" title="c14">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font>
<b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green" title="E44">A8</font></i></a>, <a class="txt" href="jordan6.html#E1:56_3_1_3"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="borsuk_1.html#D18" title="BORSUK_1:def.18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><a NAME="E6:56_3_1_3"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E7:56_3_1_3"/>
<font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E2:56_3_1_3"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="56_3_1_4"><b>case </b></a><a NAME="E1:56_3_1_4"/>
( <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K21" title="BORSUK_1:func.21">0[01]</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c2">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">g</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="borsuk_1.html#K22" title="BORSUK_1:func.22">1[01]</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c14">r1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E3:56_3_1_4"/><i><font color="Green" title="E49">A19</font></i>: 
(  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c14">r1</font> &amp; <font color="Maroon" title="c14">r1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 &amp; <font color="Maroon" title="c12">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E15:56"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="jordan6.html#E19:56"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="jordan6.html#E20:56"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="jordan6.html#E69"><i><font color="Green" title="E35">Lm1</font></i></a></i>;<br/><a NAME="E4:56_3_1_4"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default"><a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83" title="BORSUK_1:th.83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:56_3_1_4"/><b>then </b><i><font color="Green" title="E50">A20</font></i>: 
<font color="Maroon" title="c14">r1</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> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E3:56_3_1_4"><i><font color="Green" title="E49">A19</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E6:56_3_1_4"/><i><font color="Green" title="E51">A21</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P1</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P1</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="E7:56_3_1_4"/><i><font color="Green" title="E52">A22</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</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> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:56_3_1_4"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E8:56_3_1_4"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">P1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:56_3_1_4"><i><font color="Green" title="E51">A21</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon" title="c15">p</font> = <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</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/><font color="Maroon" title="c15">p</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  = 
<a href="pscomp_1.html#K13" title="PSCOMP_1:func.13">proj1</a>  <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28" title="PSCOMP_1:def.28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon" title="c2">r</font>
<b>by </b><i><a class="txt" href="jordan6.html#E3:56_3_1_4"><i><font color="Green" title="E49">A19</font></i></a>, <a class="txt" href="jordan6.html#E5:56_3_1_4"><i><font color="Green" title="E50">A20</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>
;<br/><a NAME="E11:56_3_1_4"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">r1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q</font> where <font color="Olive" title="b1">q</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> : <font color="Olive" title="b1">q</font> <a href="euclid.html#K21" title="EUCLID:func.21">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">r</font> </span>}</span> 
 ;<br/><b>hence </b><a NAME="E12:56_3_1_4"/>
<font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E6:56_3_1_4"><i><font color="Green" title="E51">A21</font></i></a>, <a class="txt" href="jordan6.html#E7:56_3_1_4"><i><font color="Green" title="E52">A22</font></i></a>, <a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E25:56"/>
( <font color="Maroon" title="c1">P1</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a>  <a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font> &amp; <font color="Maroon" title="c1">P1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" title="JORDAN6:func.6">Vertical_Line</a> <font color="Maroon" title="c2">r</font></span>)</span> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> )
 <b>by </b><i><a class="txt" href="jordan6.html#E22:56"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E23:56"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="tops_1.html#T35" title="TOPS_1:th.35">TOPS_1:35</a></i>;<br/>


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