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

<b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M1">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><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> <font color="Maroon">n</font></span>)</span>, <font color="Maroon">P2</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> <font color="Maroon">n</font></span>)</span>;<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> <font color="Maroon">n</font></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> <font color="Maroon">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green">E37</font></i>: 
( <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> &amp; <font color="Maroon">P2</font> <a href="topreal1.html#R1">is_an_arc_of</a> <font color="Maroon">p1</font>,<font color="Maroon">p2</font> &amp; <font color="Maroon">P1</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon">P1'</font> = <font color="Maroon">P1</font>, <font color="Maroon">P2'</font> = <font color="Maroon">P2</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> <font color="Maroon">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P1</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon">p</font> being    <a href="subset_1.html#M1">Element</a> of <font color="Maroon">P2</font> <a href="subset_1.html#K7">\</a> <font color="Maroon">P1</font>;<br/><a NAME="E3:51_1"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P2</font> &amp; not <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> <font color="Maroon">P1</font> )
 <b>by </b><i><a class="ref" href="jordan6.html#T3" target="_self">Th3</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon">f1</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1'</font></span>)</span><b> such that </b><br/><a NAME="E5:51_1"/><i><font color="Green">E51</font></i>: 
( <font color="Maroon">f1</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f1</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="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><a NAME="E6:51_1"/><i><font color="Green">E52</font></i>: 
<font color="Maroon">f1</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/><b>consider </b><font color="Maroon">f2</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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2'</font></span>)</span><b> such that </b><br/><a NAME="E8:51_1"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">f2</font> <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a>  &amp; <font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <font color="Maroon">p1</font> &amp; <font color="Maroon">f2</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="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="topreal1.html#D2">TOPREAL1:def 2</a></i>;<br/><b>reconsider </b><font color="Maroon">f4</font> = <font color="Maroon">f2</font> as   <a href="funct_1.html#NM1">Function</a> ;<br/><a NAME="E10:51_1"/><i><font color="Green">E55</font></i>: 
<font color="Maroon">f2</font> is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E11:51_1"/><i><font color="Green">E56</font></i>: 
(  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a>  &amp;  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> <a href="hidden.html#R1">=</a>  <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span> )
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E12:51_1"/><i><font color="Green">E57</font></i>: 
<font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a>  <a href="tops_2.html#NR1" target="_self">is_homeomorphism</a> 
 <b>by </b><i>, <a class="ref" href="tops_2.html#T70">TOPS_2:70</a></i>;<br/><b>then </b><i><font color="Green">E58</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> = 
 <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E14:51_1"/>
<font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a>  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/><b>then consider </b><font color="Maroon">h</font> being   <a href="struct_0.html#NM6">Function</a> of <a href="borsuk_1.html#K20">I[01]</a> ,<a href="borsuk_1.html#K20">I[01]</a> <b> such that </b><br/><a NAME="E16:51_1"/><i><font color="Green">E59</font></i>: 
( <font color="Maroon">h</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="partfun1.html#K1">*</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">h</font> is <a href="pre_topc.html#V5">continuous</a> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, , </i>;<br/><b>reconsider </b><font color="Maroon">h1</font> = <font color="Maroon">h</font> as   <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="funct_2.html#T9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/><a NAME="E18:51_1"/>
<font color="Maroon">h1</font> is <a href="pre_topc.html#V5">continuous</a>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:51_1_2"/>
for <font color="Olive">F1</font> being  <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>   st <font color="Olive">F1</font> is <a href="pre_topc.html#V4">closed</a> holds <br/><font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Olive">F1</font> is <a href="pre_topc.html#V4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">F1</font> be   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1"/><i><font color="Green">E60</font></i>: 
<font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon">K</font> = the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a>  as   <a href="struct_0.html#NM2">Subset</a> of <a href="topmetr.html#K3">R^1</a>  <b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:51_1_2_1"/><i><font color="Green">E78</font></i>: 
 <a href="borsuk_1.html#K20">I[01]</a>  <a href="hidden.html#R1">=</a> <a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">K</font>
 
<b>by </b><i><a class="ref" href="borsuk_1.html#T83">BORSUK_1:83</a>, <a class="ref" href="topmetr.html#T26">TOPMETR:26</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon">P3</font> = <font color="Maroon">F1</font> <a href="subset_1.html#K9">/\</a> <font color="Maroon">K</font> as   <a href="struct_0.html#NM2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3">R^1</a>  <a href="pre_topc.html#K3">|</a> <font color="Maroon">K</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T38">TOPS_2:38</a></i>;<br/>
<a NAME="E5:51_1_2_1"/><i><font color="Green">E79</font></i>: 
<font color="Maroon">P3</font> is <a href="pre_topc.html#V4">closed</a>
 
<b>by </b><i>, </i>;<br/>
<i><font color="Green">E80</font></i>: <font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">K</font> = 
the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i><a class="ref" href="funct_2.html#T48">FUNCT_2:48</a></i>
<br/>.= 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E7:51_1_2_1"/>
<font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">h1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167">RELAT_1:167</a></i>;<br/>
<b>then </b><font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> = 
<span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">K</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green">E51</font></i></a>, <a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon">h</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">P3</font>
<b>by </b><i><a class="ref" href="funct_1.html#T137">FUNCT_1:137</a></i>
;<br/>
<b>hence </b><a NAME="E9:51_1_2_1"/>
<font color="Maroon">h1</font> <a href="relset_1.html#K11">"</a> <font color="Maroon">F1</font> is <a href="pre_topc.html#V4">closed</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green">E49</font></i></a>, , <a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a>, <a class="ref" href="topmetr.html#T27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1_2"/>
<font color="Maroon">h1</font> is <a href="pre_topc.html#V5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon">g</font> = <font color="Maroon">h1</font> as  <a href="pre_topc.html#V5">continuous</a> <a href="struct_0.html#NM6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<a href="topmetr.html#K3">R^1</a>  ;<br/><i><font color="Green">E81</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<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></i>
;<br/><a NAME="E21:51_1"/><b>then </b><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">f1</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E22:51_1"/><i><font color="Green">E83</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E23:51_1"/><i><font color="Green">E84</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green">E49</font></i></a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><a NAME="E24:51_1"/><i><font color="Green">E105</font></i>: 
<font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font>
 <b>by </b><i>, <a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>;<br/><i><font color="Green">E118</font></i>:  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font> = 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<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></i>
;<br/><a NAME="E26:51_1"/><b>then </b><i><font color="Green">E119</font></i>: 
0 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</font>
 <b>by </b><i><a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><a NAME="E27:51_1"/><i><font color="Green">E120</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f2</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="E28:51_1"/><i><font color="Green">E121</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E29:51_1"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f4</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p2</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>;<br/><a NAME="E30:51_1"/><b>then </b><i><font color="Green">E122</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 0 <a href="hidden.html#R1">=</a> 0 &amp; <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> 1 )
 <b>by </b><i>, <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, <a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green">E51</font></i></a>, <a class="txt" href="jordan6.html#E6:51_1"><i><font color="Green">E52</font></i></a>, <a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green">E54</font></i></a>, <a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green">E55</font></i></a>, <a class="txt" href="jordan6.html#E11:51_1"><i><font color="Green">E56</font></i></a>, <a class="ref" href="funct_1.html#T54">FUNCT_1:54</a></i>;<br/><a NAME="E31:51_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a> 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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2'</font></span>)</span>
 <b>by </b><i>, <a class="ref" href="pre_topc.html#T29">PRE_TOPC:29</a></i>;<br/><a NAME="E32:51_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <a href="borsuk_1.html#K20">I[01]</a> 
 <b>by </b><i><a class="ref" href="funct_2.html#T7">FUNCT_2:7</a></i>;<br/><a NAME="E33:51_1"/><b>then </b><i><font color="Green">E147</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</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></i>;<br/><div><i><font color="Green">E148</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_5"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_5"/>
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">g</font>
 ;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:51_1_5"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:51_1_5"/><i><font color="Green">E152</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="funct_1.html#T22">FUNCT_1:22</a></i>;<br/><a NAME="E5:51_1_5"/><i><font color="Green">E155</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font> &amp; <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, , <a class="ref" href="funct_1.html#T21">FUNCT_1:21</a></i>;<br/><a NAME="E6:51_1_5"/>
<font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a>  is <a href="funct_1.html#V2">one-to-one</a>
 <b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>;<br/><a NAME="E7:51_1_5"/><b>then </b>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f1</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">x</font>
 <b>by </b><i>, , <a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:51_1_5"/><b>then </b><i><font color="Green">E156</font></i>: 
<font color="Maroon">p</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font>
 <b>by </b><i><a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/> <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> = 
 <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P1</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><b>hence </b><a NAME="E10:51_1_5"/>
contradiction
 <b>by </b><i><a class="ref" href="jordan6.html#T3" target="_self">Th3</a>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon">r</font> = <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font> as   <a href="real_1.html#NM1">Real</a> <b>by </b><i><a class="txt" href="jordan6.html#E13:51_1"><i><font color="Green">E58</font></i></a></i>;<br/><i><font color="Green">E157</font></i>:  <a href="relset_1.html#K5">rng</a> <font color="Maroon">f2</font> = 
 <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P2</font></span>)</span>
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon">P2</font>
<b>by </b><i><a class="ref" href="pre_topc.html#D10">PRE_TOPC:def 10</a></i>
;<br/><a NAME="E37:51_1"/><i><font color="Green">E159</font></i>: 
( 0 <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> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E13:51_1"><i><font color="Green">E58</font></i></a>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><div><i><font color="Green">E161</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_7"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_7"/><i><font color="Green">E162</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f4</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><b>hence </b><a NAME="E3:51_1_7"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, , , , <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_8"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:51_1_8"/><i><font color="Green">E164</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R1">=</a> 1
 ;<br/><font color="Maroon">f2</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">r</font> = 
<font color="Maroon">f4</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">f4</font> <a href="funct_1.html#K2">"</a> </span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">p</font></span>)</span>
<b>by </b><i><a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="tops_2.html#D4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i>, <a class="ref" href="jordan6.html#T4" target="_self">Th4</a>, , <a class="ref" href="funct_1.html#T57">FUNCT_1:57</a></i>
;<br/><b>hence </b><a NAME="E3:51_1_8"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, , , , <a class="ref" href="topreal1.html#T4">TOPREAL1:4</a></i>;<br/></div><b>end;</b></div><a NAME="E40:51_1"/><b>then </b>
( 0 <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> 1 )
 <b>by </b><i><a class="ref" href="jordan6.html#T6" target="_self">Th6</a>, <a class="ref" href="jordan6.html#T7" target="_self">Th7</a>, <a class="ref" href="real_1.html#D5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon">rc</font> being   <a href="real_1.html#NM1">Real</a><b> such that </b><br/><a NAME="E42:51_1"/><i><font color="Green">E166</font></i>: 
( <font color="Maroon">g</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">rc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">r</font> &amp; 0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">rc</font> &amp; <font color="Maroon">rc</font> <a href="xxreal_0.html#R1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E12:51_1"><i><font color="Green">E57</font></i></a>, <a class="ref" href="topreal5.html#T12">TOPREAL5:12</a></i>;<br/>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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span> = 
 <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> <font color="Maroon">n</font></span>)</span> <a href="pre_topc.html#K3">|</a> <font color="Maroon">P1</font></span>)</span>

<br/>.= 
<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="E44:51_1"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">f1</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">f2</font> <a href="tops_2.html#NK3" target="_self">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, , <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>then </b> <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font> = 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">f1</font>
<b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green">E47</font></i></a>, <a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>
<br/>.= 
 <a href="pre_topc.html#K2">[#]</a> <a href="borsuk_1.html#K20">I[01]</a> 
<b>by </b><i>, <a class="ref" href="tops_2.html#D5">TOPS_2:def 5</a></i>
<br/>.= 
<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></i>
;<br/><a NAME="E46:51_1"/><b>then </b>
<font color="Maroon">rc</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">g</font>
 <b>by </b><i>, <a class="ref" href="rcomp_1.html#T48">RCOMP_1:48</a></i>;<br/><b>hence </b><a NAME="E47:51_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green">E59</font></i></a>, , <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:51"/><b>then </b>
<font color="Maroon">P2</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E5:51"/>
<font color="Maroon">P1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">P2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green">E37</font></i></a>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/>


</div>
