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

<b>let </b><font color="Maroon" title="c1">n</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/><b>let </b><font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c3">P2</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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c4">p1</font>, <font color="Maroon" title="c5">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> <font color="Maroon" title="c1">n</font></span>)</span>;<br/>





<b>assume </b><a NAME="E1:51"/><i><font color="Green" title="E37">A1</font></i>: 
( <font color="Maroon" title="c2">P1</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c3">P2</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c4">p1</font>,<font color="Maroon" title="c5">p2</font> &amp; <font color="Maroon" title="c2">P1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c3">P2</font> )
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c6">P1'</font> = <font color="Maroon" title="c2">P1</font>, <font color="Maroon" title="c7">P2'</font> = <font color="Maroon" title="c3">P2</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> <font color="Maroon" title="c1">n</font></span>)</span> <b>by </b><i><a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">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" title="E38">A2</font></i>: 
<font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 ;<br/><b>consider </b><font color="Maroon" title="c8">p</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c3">P2</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Maroon" title="c2">P1</font>;<br/><a NAME="E3:51_1"/><i><font color="Green" title="E39">A3</font></i>: 
( <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">P2</font> &amp; not <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">P1</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/><b>consider </b><font color="Maroon" title="c9">f1</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c6">P1'</font></span>)</span><b> such that </b><br/><a NAME="E5:51_1"/><i><font color="Green" title="E40">A4</font></i>: 
( <font color="Maroon" title="c9">f1</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> &amp; <font color="Maroon" title="c9">f1</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="c5">p2</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><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="E6:51_1"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c9">f1</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#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>consider </b><font color="Maroon" title="c10">f2</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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P2'</font></span>)</span><b> such that </b><br/><a NAME="E8:51_1"/><i><font color="Green" title="E42">A6</font></i>: 
( <font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p1</font> &amp; <font color="Maroon" title="c10">f2</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="c5">p2</font> )
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><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/><b>reconsider </b><font color="Maroon" title="c11">f4</font> = <font color="Maroon" title="c10">f2</font> as   <a href="funct_1.html#NM1" title="FUNCT_1:NM.1">Function</a> ;<br/><a NAME="E10:51_1"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c10">f2</font> is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><a NAME="E11:51_1"/><i><font color="Green" title="E44">A8</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</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="E12:51_1"/><i><font color="Green" title="E45">A9</font></i>: 
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a> 
 <b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_2.html#T70" title="TOPS_2:th.70">TOPS_2:70</a></i>;<br/><b>then </b><i><font color="Green" title="E46">A10</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> = 
 <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span>
<b>by </b><i><a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">P2</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="E14:51_1"/>
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c12">h</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> ,<a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> <b> such that </b><br/><a NAME="E16:51_1"/><i><font color="Green" title="E47">A11</font></i>: 
( <font color="Maroon" title="c12">h</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="partfun1.html#K1" title="PARTFUN1:func.1">*</a> <font color="Maroon" title="c9">f1</font> &amp; <font color="Maroon" title="c12">h</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#E1:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E6:51_1"><i><font color="Green" title="E41">A5</font></i></a>, <a class="ref" href="jordan6.html#T58" target="_self" title="JORDAN6:th.58">Th58</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c13">h1</font> = <font color="Maroon" title="c12">h</font> as   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <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#T9" title="FUNCT_2:th.9">FUNCT_2:9</a>, <a class="ref" href="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/><a NAME="E18:51_1"/>
<font color="Maroon" title="c13">h1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">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" title="b1">F1</font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>   st <font color="Olive" title="b1">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a> holds <br/><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Olive" title="b1">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="51_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c14">F1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a> ;<br/>

<b>assume </b><a NAME="E1:51_1_2_1"/><i><font color="Green" title="E48">A12</font></i>: 
<font color="Maroon" title="c14">F1</font> is <a href="pre_topc.html#V4" title="PRE_TOPC:attr.4">closed</a>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c15">K</font> = 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>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <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="topmetr.html#T24" title="TOPMETR:th.24">TOPMETR:24</a></i>;<br/>
<a NAME="E3:51_1_2_1"/><i><font color="Green" title="E49">A13</font></i>: 
 <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">K</font>
 
<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="topmetr.html#T26" title="TOPMETR:th.26">TOPMETR:26</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c16">P3</font> = <font color="Maroon" title="c14">F1</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c15">K</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c15">K</font></span>)</span> <b>by </b><i><a class="ref" href="tops_2.html#T38" title="TOPS_2:th.38">TOPS_2:38</a></i>;<br/>
<a NAME="E5:51_1_2_1"/><i><font color="Green" title="E50">A14</font></i>: 
<font color="Maroon" title="c16">P3</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#E1:51_1_2_1"><i><font color="Green" title="E48">A12</font></i></a>, <a class="ref" href="jordan6.html#T3" target="_self" title="JORDAN6:th.3">Th3</a></i>;<br/>
<i><font color="Green" title="E51">A15</font></i>: <font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c15">K</font> = 
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> 
<b>by </b><i><a class="ref" href="funct_2.html#T48" title="FUNCT_2:th.48">FUNCT_2:48</a></i>
<br/>.= 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h1</font>
<b>by </b><i><a class="ref" href="funct_2.html#D1" title="FUNCT_2:def.1">FUNCT_2:def 1</a></i>
;<br/>
<a NAME="E7:51_1_2_1"/>
<font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c14">F1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c13">h1</font>
 
<b>by </b><i><a class="ref" href="relat_1.html#T167" title="RELAT_1:th.167">RELAT_1:167</a></i>;<br/>
<b>then </b><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c14">F1</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c14">F1</font></span>)</span> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <span class="p1">(<span class="default"><font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c15">K</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E6:51_1_2_1"><i><font color="Green" title="E51">A15</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>
<br/>.= 
<font color="Maroon" title="c12">h</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c16">P3</font>
<b>by </b><i><a class="ref" href="funct_1.html#T137" title="FUNCT_1:th.137">FUNCT_1:137</a></i>
;<br/>
<b>hence </b><a NAME="E9:51_1_2_1"/>
<font color="Maroon" title="c13">h1</font> <a href="relset_1.html#K11" title="RELSET_1:func.11">"</a> <font color="Maroon" title="c14">F1</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#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1_2_1"><i><font color="Green" title="E49">A13</font></i></a>, <a class="txt" href="jordan6.html#E5:51_1_2_1"><i><font color="Green" title="E50">A14</font></i></a>, <a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:51_1_2"/>
<font color="Maroon" title="c13">h1</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a>
 <b>by </b><i><a class="ref" href="pre_topc.html#D12" title="PRE_TOPC:def.12">PRE_TOPC:def 12</a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c14">g</font> = <font color="Maroon" title="c13">h1</font> as  <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <span class="p1">(<span class="default"><a href="topmetr.html#K4" title="TOPMETR:func.4">Closed-Interval-TSpace</a> 0,1</span>)</span>,<a href="topmetr.html#K3" title="TOPMETR:func.3">R^1</a>  ;<br/><i><font color="Green" title="E48">A16</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
<b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</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></i>
;<br/><a NAME="E21:51_1"/><b>then </b><i><font color="Green" title="E49">A17</font></i>: 
0 <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="c9">f1</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="E22:51_1"/><i><font color="Green" title="E50">A18</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="c9">f1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E20:51_1"><i><font color="Green" title="E48">A16</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E23:51_1"/><i><font color="Green" title="E51">A19</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E21:51_1"><i><font color="Green" title="E49">A17</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><a NAME="E24:51_1"/><i><font color="Green" title="E52">A20</font></i>: 
<font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E22:51_1"><i><font color="Green" title="E50">A18</font></i></a>, <a class="ref" href="funct_1.html#T23" title="FUNCT_1:th.23">FUNCT_1:23</a></i>;<br/><i><font color="Green" title="E53">A21</font></i>:  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c10">f2</font> = 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
<b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</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></i>
;<br/><a NAME="E26:51_1"/><b>then </b><i><font color="Green" title="E54">A22</font></i>: 
0 <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="c10">f2</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="E27:51_1"/><i><font color="Green" title="E55">A23</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="c10">f2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E25:51_1"><i><font color="Green" title="E53">A21</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E28:51_1"/><i><font color="Green" title="E56">A24</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c4">p1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E11:51_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E29:51_1"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c5">p2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E11:51_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>;<br/><a NAME="E30:51_1"/><b>then </b><i><font color="Green" title="E57">A25</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0 &amp; <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E23:51_1"><i><font color="Green" title="E51">A19</font></i></a>, <a class="txt" href="jordan6.html#E24:51_1"><i><font color="Green" title="E52">A20</font></i></a>, <a class="txt" href="jordan6.html#E26:51_1"><i><font color="Green" title="E54">A22</font></i></a>, <a class="txt" href="jordan6.html#E27:51_1"><i><font color="Green" title="E55">A23</font></i></a>, <a class="txt" href="jordan6.html#E28:51_1"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="funct_1.html#T54" title="FUNCT_1:th.54">FUNCT_1:54</a></i>;<br/><a NAME="E31:51_1"/>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c7">P2'</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="ref" href="pre_topc.html#T29" title="PRE_TOPC:th.29">PRE_TOPC:29</a></i>;<br/><a NAME="E32:51_1"/><b>then </b>
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> 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> 
 <b>by </b><i><a class="ref" href="funct_2.html#T7" title="FUNCT_2:th.7">FUNCT_2:7</a></i>;<br/><a NAME="E33:51_1"/><b>then </b><i><font color="Green" title="E58">A26</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</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></i>;<br/><div><i><font color="Green" title="E59">A27</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" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</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="c14">g</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c15">x</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:51_1_5"/><i><font color="Green" title="E60">A28</font></i>: 
( <font color="Maroon" title="c15">x</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="c14">g</font> &amp; <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> )
 <b>by </b><i><a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/><a NAME="E4:51_1_5"/><i><font color="Green" title="E61">A29</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1_5"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="funct_1.html#T22" title="FUNCT_1:th.22">FUNCT_1:22</a></i>;<br/><a NAME="E5:51_1_5"/><i><font color="Green" title="E62">A30</font></i>: 
( <font color="Maroon" title="c15">x</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="c9">f1</font> &amp; <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> )
 <b>by </b><i><a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1_5"><i><font color="Green" title="E60">A28</font></i></a>, <a class="ref" href="funct_1.html#T21" title="FUNCT_1:th.21">FUNCT_1:21</a></i>;<br/><a NAME="E6:51_1_5"/>
<font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a>  is <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>
 <b>by </b><i><a class="txt" href="jordan6.html#E12:51_1"><i><font color="Green" title="E45">A9</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="E7:51_1_5"/><b>then </b>
<font color="Maroon" title="c8">p</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">f1</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">x</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E13:51_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="txt" href="jordan6.html#E4:51_1_5"><i><font color="Green" title="E61">A29</font></i></a>, <a class="txt" href="jordan6.html#E5:51_1_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/><a NAME="E8:51_1_5"/><b>then </b><i><font color="Green" title="E63">A31</font></i>: 
<font color="Maroon" title="c8">p</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="c9">f1</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E5:51_1_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/> <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f1</font> = 
 <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c2">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>hence </b><a NAME="E10:51_1_5"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51_1"><i><font color="Green" title="E38">A2</font></i></a>, <a class="txt" href="jordan6.html#E8:51_1_5"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><b>reconsider </b><font color="Maroon" title="c15">r</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan6.html#E33:51_1"><i><font color="Green" title="E58">A26</font></i></a></i>;<br/><i><font color="Green" title="E60">A32</font></i>:  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c10">f2</font> = 
 <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c3">P2</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<font color="Maroon" title="c3">P2</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="E37:51_1"/><i><font color="Green" title="E61">A33</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c15">r</font> &amp; <font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E33:51_1"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><div><i><font color="Green" title="E62">A34</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" title="E63">A35</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 0
 ;<br/><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">r</font> = 
<font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E11:51_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E36:51_1"><i><font color="Green" title="E60">A32</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">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" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="jordan6.html#E1:51_1_7"><i><font color="Green" title="E63">A35</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">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" title="E63">A36</font></i>: 
<font color="Maroon" title="c15">r</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1
 ;<br/><font color="Maroon" title="c10">f2</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c15">r</font> = 
<font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c11">f4</font> <a href="funct_1.html#K2" title="FUNCT_1:func.2">"</a> </span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c8">p</font></span>)</span>
<b>by </b><i><a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E11:51_1"><i><font color="Green" title="E44">A8</font></i></a>, <a class="ref" href="tops_2.html#D4" title="TOPS_2:def.4">TOPS_2:def 4</a></i>
<br/>.= 
<font color="Maroon" title="c8">p</font>
<b>by </b><i><a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E10:51_1"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="jordan6.html#E36:51_1"><i><font color="Green" title="E60">A32</font></i></a>, <a class="ref" href="funct_1.html#T57" title="FUNCT_1:th.57">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" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E3:51_1"><i><font color="Green" title="E39">A3</font></i></a>, <a class="txt" href="jordan6.html#E8:51_1"><i><font color="Green" title="E42">A6</font></i></a>, <a class="txt" href="jordan6.html#E1:51_1_8"><i><font color="Green" title="E63">A36</font></i></a>, <a class="ref" href="topreal1.html#T4" title="TOPREAL1:th.4">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" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c15">r</font> &amp; <font color="Maroon" title="c15">r</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E37:51_1"><i><font color="Green" title="E61">A33</font></i></a>, <a class="txt" href="jordan6.html#E38:51_1"><i><font color="Green" title="E62">A34</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><b>then consider </b><font color="Maroon" title="c16">rc</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E42:51_1"/><i><font color="Green" title="E63">A37</font></i>: 
( <font color="Maroon" title="c14">g</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">rc</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c15">r</font> &amp; 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">rc</font> &amp; <font color="Maroon" title="c16">rc</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1 )
 <b>by </b><i><a class="txt" href="jordan6.html#E30:51_1"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="topreal5.html#T12" title="TOPREAL5:th.12">TOPREAL5:12</a></i>;<br/>the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span> = 
 <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> <font color="Maroon" title="c1">n</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">P1</font></span>)</span>

<br/>.= 
<font color="Maroon" title="c2">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="E44:51_1"/><b>then </b>
 <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c9">f1</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c10">f2</font> <a href="tops_2.html#NK3" title="TOPS_2:NK.3">"</a> </span>)</span>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="txt" href="jordan6.html#E13:51_1"><i><font color="Green" title="E46">A10</font></i></a>, <a class="ref" href="xboole_1.html#T1" title="XBOOLE_1:th.1">XBOOLE_1:1</a></i>;<br/><b>then </b> <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c14">g</font> = 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c9">f1</font>
<b>by </b><i><a class="txt" href="jordan6.html#E16:51_1"><i><font color="Green" title="E47">A11</font></i></a>, <a class="ref" href="relat_1.html#T46" title="RELAT_1:th.46">RELAT_1:46</a></i>
<br/>.= 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="borsuk_1.html#K20" title="BORSUK_1:func.20">I[01]</a> 
<b>by </b><i><a class="txt" href="jordan6.html#E5:51_1"><i><font color="Green" title="E40">A4</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>
<br/>.= 
<span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</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></i>
;<br/><a NAME="E46:51_1"/><b>then </b>
<font color="Maroon" title="c16">rc</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="c14">g</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E42:51_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">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#E34:51_1"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="jordan6.html#E42:51_1"><i><font color="Green" title="E63">A37</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/></div><b>end;</b></div>
<a NAME="E4:51"/><b>then </b>
<font color="Maroon" title="c3">P2</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">P1</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T37" title="XBOOLE_1:th.37">XBOOLE_1:37</a></i>;<br/>
<b>hence </b><a NAME="E5:51"/>
<font color="Maroon" title="c2">P1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">P2</font>
 <b>by </b><i><a class="txt" href="jordan6.html#E1:51"><i><font color="Green" title="E37">A1</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>


</div>
