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

<b>let </b><font color="Maroon">P1</font> be   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon">r</font> be  <a href="xreal_0.html#V1">real</a>  <a href="ordinal1.html#NM1" target="_self">number</a> ;<br/><b>let </b><font color="Maroon">p1</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <font color="Maroon">p2</font> be   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:56"/><i><font color="Green">E37</font></i>: 
( <font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a>  <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a>  &amp; <font color="Maroon">P1</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">P1'</font> = <font color="Maroon">P1</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<b>consider </b><font color="Maroon">f</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1'</font></span>)</span><b> such that </b><br/><a NAME="E4:56"/><i><font color="Green">E47</font></i>: 
( <font color="Maroon">f</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">p2</font> )
 <b>by </b><i>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E5:56"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E6:56"/><b>then </b><i><font color="Green">E49</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon">f1</font> = <font color="Maroon">f</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="ref" href="funct_2.html#T9">FUNCT_2:9</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f2</font> = <font color="Maroon">f1</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">proj11</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>, <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">proj12</font> =  <a href="pscomp_1.html#K13">proj1</a>  as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<b>reconsider </b><font color="Maroon">g1</font> = <font color="Maroon">proj11</font> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> as   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> , <a href="numbers.html#K1">REAL</a>  ;<br/>
<b>reconsider </b><font color="Maroon">g</font> = <font color="Maroon">g1</font> as   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E13:56"/>
<font color="Maroon">f</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E14:56"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">f2</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i/>;<br/>
<a NAME="E15:56"/>
<font color="Maroon">proj12</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i><a class="ref" href="topreal5.html#T16">TOPREAL5:16</a></i>;<br/>
<a NAME="E16:56"/><b>then </b><i><font color="Green">E52</font></i>: 
<font color="Maroon">g</font> is <a href="pre_topc.html#V5">continuous</a>
 
<b>by </b><i>, <a class="ref" href="tops_2.html#T58">TOPS_2:58</a></i>;<br/>
<a NAME="E17:56"/><i><font color="Green">E54</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E18:56"/><b>then </b><i><font color="Green">E55</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<a NAME="E19:56"/><i><font color="Green">E56</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 
<b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/>
<i><font color="Green">E57</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p1</font>
<b>by </b><i>, <a class="txt" href="jordan6.html#E4:56"><i><font color="Green">E47</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">p1</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/>
<i><font color="Green">E58</font></i>: <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_2.html#K3">.</a> <font color="Maroon">p2</font>
<b>by </b><i>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
<br/>.= 
<font color="Maroon">p2</font> <a href="euclid.html#K21">`1</a> 
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">P1'</font> = <font color="Maroon">P1'</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E23:56"/><i><font color="Green">E59</font></i>: 
<font color="Maroon">P1'</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<a NAME="E24:56"/><i><font color="Green">E60</font></i>: 
 <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><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">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 or <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> or ( <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  ) )
 </span><b>by </b><i>, <a class="txt" href="jordan6.html#E6:56"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</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">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1
 ;<br/><div class="add"><a NAME="E2:56_3_1_1"/><b>then </b><i><font color="Green">E78</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E6:56"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E3:56_3_1_1"/><i><font color="Green">E79</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E4:56"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E4:56_3_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E5:56_3_1_1"/><b>then </b><i><font color="Green">E80</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E17:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E8:56_3_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><a NAME="E9:56_3_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><a NAME="E10:56_3_1_1"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#T3">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">E81</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:56_3_1_2"/><i><font color="Green">E82</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E4:56"><i><font color="Green">E47</font></i></a></i>;<br/><a NAME="E3:56_3_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:56_3_1_2"/><b>then </b><i><font color="Green">E83</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <font color="Maroon">P1'</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0</span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E17:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="borsuk_1.html#D17">BORSUK_1:def 17</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E7:56_3_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><a NAME="E8:56_3_1_2"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><a NAME="E9:56_3_1_2"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#T3">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">E84</font></i>: 
<font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font>
 ;<br/><div class="add"><a NAME="E2:56_3_1_3"/><i><font color="Green">E105</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i/>;<br/><a NAME="E3:56_3_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:56_3_1_3"/><b>then </b><i><font color="Green">E118</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E17:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="borsuk_1.html#D18">BORSUK_1:def 18</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E7:56_3_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><a NAME="E8:56_3_1_3"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><a NAME="E9:56_3_1_3"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green">E55</font></i></a>, <a class="ref" href="xboole_0.html#T3">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">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K21">0[01]</a>  <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r</font> &amp; <font color="Maroon">r</font> <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">g</font> <a href="funct_2.html#K3">.</a> <a href="borsuk_1.html#K22">1[01]</a>  )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">r1</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E3:56_3_1_4"/><i><font color="Green">E119</font></i>: 
( 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">r1</font> &amp; <font color="Maroon">r1</font> <a href="xxreal_0.html#R1">&lt;</a> 1 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> )
 <b>by </b><i>, <a class="txt" href="jordan6.html#E6:56"><i><font color="Green">E49</font></i></a>, , </i>;<br/><a NAME="E4:56_3_1_4"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="rcomp_1.html#K1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1">.]</a></span>
 <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E5:56_3_1_4"/><b>then </b><i><font color="Green">E120</font></i>: 
<font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E6:56_3_1_4"/>
 <a href="pre_topc.html#K2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 <b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>;<br/><a NAME="E7:56_3_1_4"/><b>then </b><i><font color="Green">E121</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">P1</font>
 ;<br/><a NAME="E8:56_3_1_4"/>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E17:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E9:56_3_1_4"/><b>then </b><i><font color="Green">E122</font></i>: 
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1'</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E18:56"><i><font color="Green">E55</font></i></a></i>;<br/><b>then reconsider </b><font color="Maroon">p</font> = <font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> as   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> ;<br/><font color="Maroon">p</font> <a href="euclid.html#K21">`1</a>  = 
<a href="pscomp_1.html#K13">proj1</a>  <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font></span>)</span>
<b>by </b><i><a class="ref" href="pscomp_1.html#D28">PSCOMP_1:def 28</a></i>
<br/>.= 
<font color="Maroon">r</font>
<b>by </b><i><a class="ref" href="jordan6.html#T5" target="_self">Th5</a>, <a class="txt" href="jordan6.html#E17:56"><i><font color="Green">E54</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/><a NAME="E12:56_3_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <font color="Olive">q</font> where <font color="Olive">q</font> is   <a href="pre_topc.html#NM2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15">TOP-REAL</a> 2</span>)</span> : <font color="Olive">q</font> <a href="euclid.html#K21">`1</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> </span>}</span> 
 ;<br/><a NAME="E13:56_3_1_4"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r1</font> <a href="hidden.html#R2">in</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 ;<br/><b>hence </b><a NAME="E14:56_3_1_4"/>
<font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E19:56"><i><font color="Green">E56</font></i></a>, <a class="ref" href="xboole_0.html#T3">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="E26:56"/>
( <font color="Maroon">P1</font> <a href="xboole_0.html#NR5" target="_self">meets</a>  <a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font> &amp; <font color="Maroon">P1</font> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="jordan6.html#K6" target="_self">Vertical_Line</a> <font color="Maroon">r</font></span>)</span> is <a href="pre_topc.html#V4">closed</a> )
 <b>by </b><i><a class="txt" href="jordan6.html#E14:56"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E16:56"><i><font color="Green">E52</font></i></a>, <a class="ref" href="tops_1.html#T35">TOPS_1:35</a></i>;<br/>


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