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

<b>let </b><font color="Maroon" title="c1">P</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/><b>let </b><font color="Maroon" title="c2">Q</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>;<br/><b>let </b><font color="Maroon" title="c3">p1</font>, <font color="Maroon" title="c4">p2</font>, <font color="Maroon" title="c5">q</font> be   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span>;<br/>





<b>assume </b><a NAME="E1:19"/><i><font color="Green" title="E20">A1</font></i>: 
( <font color="Maroon" title="c1">P</font> <a href="topreal1.html#R1" title="TOPREAL1:pred.1">is_an_arc_of</a> <font color="Maroon" title="c3">p1</font>,<font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c5">q</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c4">p2</font> &amp; <font color="Maroon" title="c2">Q</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">f</font> being   <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span><b> such that </b><br/><a NAME="E3:19"/><i><font color="Green" title="E21">A2</font></i>: 
( <font color="Maroon" title="c6">f</font> <a href="tops_2.html#NR1" title="TOPS_2:NR.1">is_homeomorphism</a>  &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 0 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">p1</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="ref" href="topreal1.html#D2" title="TOPREAL1:def.2">TOPREAL1:def 2</a></i>;<br/>
<a NAME="E4:19"/><i><font color="Green" title="E22">A3</font></i>: 
(  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  &amp;  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/>
<a NAME="E5:19"/><i><font color="Green" title="E23">A4</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c7">xs</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E7:19"/><i><font color="Green" title="E24">A5</font></i>: 
( <font color="Maroon" title="c7">xs</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c7">xs</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<a NAME="E8:19"/><i><font color="Green" title="E25">A6</font></i>: 
 <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">[.</a><span class="default">0,1</span><a href="rcomp_1.html#K1" title="RCOMP_1:func.1">.]</a></span>
 
<b>by </b><i><a class="ref" href="topmetr.html#T25" title="TOPMETR:th.25">TOPMETR:25</a>, <a class="ref" href="topmetr.html#T27" title="TOPMETR:th.27">TOPMETR:27</a></i>;<br/>
<b>then reconsider </b><font color="Maroon" title="c8">s</font> = <font color="Maroon" title="c7">xs</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E10:19"/><i><font color="Green" title="E26">A7</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E11:19"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_1"/>
<font color="Maroon" title="c9">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">q0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_1"/><i><font color="Green" title="E27">A8</font></i>: 
( <font color="Maroon" title="c10">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">z</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c10">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c11">ss</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:19_1"/><i><font color="Green" title="E28">A9</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c11">ss</font> &amp; <font color="Maroon" title="c11">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c10">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c11">ss</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>
<a NAME="E6:19_1"/>
<font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:19_1"/><b>then </b>
<font color="Maroon" title="c11">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:19_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E8:19_1"/><b>then </b>
<font color="Maroon" title="c11">ss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E9:19_1"/><b>then </b>
<font color="Maroon" title="c10">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:19_1"><i><font color="Green" title="E28">A9</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E10:19_1"/>
<font color="Maroon" title="c9">z</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_1"><i><font color="Green" title="E27">A8</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c9">P1'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> ;<br/>
<a NAME="E13:19"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c10">z</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_2"/>
<font color="Maroon" title="c10">z</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c11">q0</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_2"/><i><font color="Green" title="E27">A10</font></i>: 
( <font color="Maroon" title="c11">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c11">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c12">ss</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:19_2"/><i><font color="Green" title="E28">A11</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c12">ss</font> &amp; <font color="Maroon" title="c12">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c11">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">ss</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>
<a NAME="E6:19_2"/>
<font color="Maroon" title="c12">ss</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:19_2"/><b>then </b>
<font color="Maroon" title="c12">ss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:19_2"/><b>then </b>
<font color="Maroon" title="c11">q0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K5" title="RELSET_1:func.5">rng</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E5:19_2"><i><font color="Green" title="E28">A11</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>hence </b><a NAME="E9:19_2"/>
<font color="Maroon" title="c10">z</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> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span>
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_2"><i><font color="Green" title="E27">A10</font></i></a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c10">P2'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span>  as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> ;<br/>
<a NAME="E15:19"/><i><font color="Green" title="E27">A12</font></i>: 
<font color="Maroon" title="c9">P1'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="jordan20.html#T15" target="_self" title="JORDAN20:th.15">Th15</a></i>;<br/>
<a NAME="E16:19"/><i><font color="Green" title="E28">A13</font></i>: 
<font color="Maroon" title="c10">P2'</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="jordan20.html#T16" target="_self" title="JORDAN20:th.16">Th16</a></i>;<br/>
<a NAME="E17:19"/><i><font color="Green" title="E29">A14</font></i>: 
<font color="Maroon" title="c2">Q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P2'</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_3"/><i><font color="Green" title="E30">A15</font></i>: 
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 ;<br/>

<a NAME="E2:19_3"/><b>then </b><i><font color="Green" title="E31">A16</font></i>: 
( <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font> &amp; not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span> )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="ref" href="xboole_0.html#D4" title="XBOOLE_0:def.4">XBOOLE_0:def 4</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c12">xt</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E4:19_3"/><i><font color="Green" title="E32">A17</font></i>: 
( <font color="Maroon" title="c12">xt</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c12">xt</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c13">t</font> = <font color="Maroon" title="c12">xt</font> as   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> <b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E4:19_3"><i><font color="Green" title="E32">A17</font></i></a></i>;<br/>
<a NAME="E6:19_3"/><i><font color="Green" title="E33">A18</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">t</font> &amp; <font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E4:19_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">qq</font> = <font color="Maroon" title="c11">x</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E1:19_3"><i><font color="Green" title="E30">A15</font></i></a></i>;<br/>
<a NAME="E8:19_3"/><i><font color="Green" title="E34">A19</font></i>: 
not <font color="Maroon" title="c11">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E2:19_3"><i><font color="Green" title="E31">A16</font></i></a>, <a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_3_1"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:19_3_1_1"/>
( <font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> or <font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">s</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_1"><b>case </b></a><a NAME="E1:19_3_1_1_1"/>
<font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font>
 ;<br/><div class="add"><a NAME="E2:19_3_1_1_1"/><b>then </b>
( <font color="Maroon" title="c14">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c14">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:19_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="jordan20.html#E6:19_3"><i><font color="Green" title="E33">A18</font></i></a></i>;<br/><a NAME="E3:19_3_1_1_1"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font>
 ;<br/><b>hence </b><a NAME="E4:19_3_1_1_1"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P2'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="19_3_1_1_2"><b>case </b></a><a NAME="E1:19_3_1_1_2"/>
<font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <font color="Maroon" title="c8">s</font>
 ;<br/><div class="add"><a NAME="E2:19_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c13">t</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">s</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E4:19_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="jordan20.html#E8:19_3"><i><font color="Green" title="E34">A19</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/><a NAME="E3:19_3_1_1_2"/><b>then </b>
( <font color="Maroon" title="c14">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c14">qq</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E4:19_3"><i><font color="Green" title="E32">A17</font></i></a>, <a class="txt" href="jordan20.html#E6:19_3"><i><font color="Green" title="E33">A18</font></i></a></i>;<br/><a NAME="E4:19_3_1_1_2"/><b>then </b>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P2'</font>
 ;<br/><b>hence </b><a NAME="E5:19_3_1_1_2"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P2'</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>
<b>hence </b><a NAME="E10:19_3"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c10">P2'</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E18:19"/>
<font color="Maroon" title="c8">s</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a></i>;<br/>
<a NAME="E19:19"/><b>then </b><i><font color="Green" title="E30">A20</font></i>: 
<font color="Maroon" title="c3">p1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a></i>;<br/>
<a NAME="E20:19"/><i><font color="Green" title="E31">A21</font></i>: 
<font color="Maroon" title="c9">P1'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">q00</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_4"/><i><font color="Green" title="E32">A22</font></i>: 
( <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">ss</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:19_4"/><i><font color="Green" title="E33">A23</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c13">ss</font> &amp; <font color="Maroon" title="c13">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ss</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_4"><i><font color="Green" title="E32">A22</font></i></a></i>;<br/>
<a NAME="E6:19_4"/>
<font color="Maroon" title="c13">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 
<b>by </b><i><a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="jordan20.html#E5:19_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/>
<a NAME="E7:19_4"/><b>then </b><i><font color="Green" title="E34">A24</font></i>: 
<font color="Maroon" title="c13">ss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:19_4"/><b>then </b><i><font color="Green" title="E35">A25</font></i>: 
<font color="Maroon" title="c12">q00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="jordan20.html#E5:19_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_4_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_4_1"/><i><font color="Green" title="E36">A26</font></i>: 
<font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 ;<br/><a NAME="E2:19_4_1"/>
<font color="Maroon" title="c6">f</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="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>hence </b><a NAME="E3:19_4_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E5:19_4"><i><font color="Green" title="E33">A23</font></i></a>, <a class="txt" href="jordan20.html#E7:19_4"><i><font color="Green" title="E34">A24</font></i></a>, <a class="txt" href="jordan20.html#E1:19_4_1"><i><font color="Green" title="E36">A26</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:19_4"/><b>then </b>
not <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:19_4"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:19_4"><i><font color="Green" title="E32">A22</font></i></a>, <a class="txt" href="jordan20.html#E8:19_4"><i><font color="Green" title="E35">A25</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>
<a NAME="E21:19"/><b>then </b>
<font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</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> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E19:19"><i><font color="Green" title="E30">A20</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E22:19"/><b>then </b><i><font color="Green" title="E32">A27</font></i>: 
<font color="Maroon" title="c9">P1'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E23:19"/>
( 1 <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c4">p2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 )
 
<b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="ref" href="real_1.html#D5" title="REAL_1:def.5">REAL_1:def 5</a></i>;<br/>
<a NAME="E24:19"/><b>then </b><i><font color="Green" title="E33">A28</font></i>: 
<font color="Maroon" title="c4">p2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">q0</font> where <font color="Olive" title="b1">q0</font> is   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> :  ex <font color="Olive" title="b2">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b2">ss</font> &amp; <font color="Olive" title="b2">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Olive" title="b1">q0</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b2">ss</font> ) </span>}</span> 
 
;<br/>
<a NAME="E25:19"/><i><font color="Green" title="E34">A29</font></i>: 
<font color="Maroon" title="c10">P2'</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">Q</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c11">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:19_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P2'</font>
 ;<br/>

<b>then consider </b><font color="Maroon" title="c12">q00</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E3:19_5"/><i><font color="Green" title="E35">A30</font></i>: 
( <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">x</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 ;<br/>
<b>consider </b><font color="Maroon" title="c13">ss</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E5:19_5"/><i><font color="Green" title="E36">A31</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c13">ss</font> &amp; <font color="Maroon" title="c13">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c13">ss</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_5"><i><font color="Green" title="E35">A30</font></i></a></i>;<br/>
<a NAME="E6:19_5"/>
<font color="Maroon" title="c13">ss</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 
<b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_5"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E7:19_5"/><b>then </b><i><font color="Green" title="E37">A32</font></i>: 
<font color="Maroon" title="c13">ss</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E5:19_5"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/>
<a NAME="E8:19_5"/><b>then </b><i><font color="Green" title="E38">A33</font></i>: 
<font color="Maroon" title="c12">q00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c1">P</font>
 
<b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E5:19"><i><font color="Green" title="E23">A4</font></i></a>, <a class="txt" href="jordan20.html#E5:19_5"><i><font color="Green" title="E36">A31</font></i></a>, <a class="ref" href="funct_1.html#D5" title="FUNCT_1:def.5">FUNCT_1:def 5</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="19_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_5_1"/><i><font color="Green" title="E39">A34</font></i>: 
<font color="Maroon" title="c12">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">q</font>
 ;<br/><a NAME="E2:19_5_1"/>
<font color="Maroon" title="c6">f</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="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>hence </b><a NAME="E3:19_5_1"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E5:19_5"><i><font color="Green" title="E36">A31</font></i></a>, <a class="txt" href="jordan20.html#E7:19_5"><i><font color="Green" title="E37">A32</font></i></a>, <a class="txt" href="jordan20.html#E1:19_5_1"><i><font color="Green" title="E39">A34</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<a NAME="E10:19_5"/><b>then </b>
not <font color="Maroon" title="c12">q00</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="tarski.html#K1" title="TARSKI:func.1">{</a><span class="default"><font color="Maroon" title="c5">q</font></span><a href="tarski.html#K1" title="TARSKI:func.1">}</a></span>
 
<b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E11:19_5"/>
<font color="Maroon" title="c11">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">Q</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E1:19"><i><font color="Green" title="E20">A1</font></i></a>, <a class="txt" href="jordan20.html#E3:19_5"><i><font color="Green" title="E35">A30</font></i></a>, <a class="txt" href="jordan20.html#E8:19_5"><i><font color="Green" title="E38">A33</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>
<a NAME="E26:19"/><b>then </b>
<font color="Maroon" title="c10">P2'</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c2">Q</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> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E24:19"><i><font color="Green" title="E33">A28</font></i></a>, <a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E27:19"/><b>then </b><i><font color="Green" title="E35">A35</font></i>: 
<font color="Maroon" title="c10">P2'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c2">Q</font>
 
<b>by </b><i><a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c11">Q'</font> = <font color="Maroon" title="c2">Q</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"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <b>by </b><i><a class="txt" href="jordan20.html#E24:19"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="jordan20.html#E25:19"><i><font color="Green" title="E34">A29</font></i></a></i>;<br/>
<div><i><font color="Green" title="E36">A36</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="19_6"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:19_6"/>
<font color="Maroon" title="c9">P1'</font> <a href="xboole_0.html#NR5" title="XBOOLE_0:NR.5">meets</a> <font color="Maroon" title="c10">P2'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">p0</font> being    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> <b> such that </b><br/><a NAME="E3:19_6"/><i><font color="Green" title="E37">A37</font></i>: 
( <font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">P1'</font> &amp; <font color="Maroon" title="c12">p0</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c10">P2'</font> )
 <b>by </b><i><a class="ref" href="xboole_0.html#T3" title="XBOOLE_0:th.3">XBOOLE_0:3</a></i>;<br/><b>consider </b><font color="Maroon" title="c13">q00</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E5:19_6"/><i><font color="Green" title="E38">A38</font></i>: 
( <font color="Maroon" title="c13">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p0</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c13">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_6"><i><font color="Green" title="E37">A37</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c14">ss1</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E7:19_6"/><i><font color="Green" title="E39">A39</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c14">ss1</font> &amp; <font color="Maroon" title="c14">ss1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c8">s</font> &amp; <font color="Maroon" title="c13">q00</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c14">ss1</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E5:19_6"><i><font color="Green" title="E38">A38</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c15">q01</font> being   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span><b> such that </b><br/><a NAME="E9:19_6"/><i><font color="Green" title="E40">A40</font></i>: 
( <font color="Maroon" title="c15">q01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c12">p0</font> &amp;  ex <font color="Olive" title="b1">ss</font> being  <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a> st <br/>( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Olive" title="b1">ss</font> &amp; <font color="Olive" title="b1">ss</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">q01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">ss</font> ) )
 <b>by </b><i><a class="txt" href="jordan20.html#E3:19_6"><i><font color="Green" title="E37">A37</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c16">ss2</font> being   <a href="real_1.html#NM1" title="REAL_1:NM.1">Real</a><b> such that </b><br/><a NAME="E11:19_6"/><i><font color="Green" title="E41">A41</font></i>: 
( <font color="Maroon" title="c8">s</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> <font color="Maroon" title="c16">ss2</font> &amp; <font color="Maroon" title="c16">ss2</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> 1 &amp; <font color="Maroon" title="c15">q01</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c16">ss2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E9:19_6"><i><font color="Green" title="E40">A40</font></i></a></i>;<br/><a NAME="E12:19_6"/>
<font color="Maroon" title="c14">ss1</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;</a> 1
 <b>by </b><i><a class="txt" href="jordan20.html#E10:19"><i><font color="Green" title="E26">A7</font></i></a>, <a class="txt" href="jordan20.html#E7:19_6"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="xxreal_0.html#T2" title="XXREAL_0:th.2">XXREAL_0:2</a></i>;<br/><a NAME="E13:19_6"/><b>then </b><i><font color="Green" title="E42">A42</font></i>: 
<font color="Maroon" title="c14">ss1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E7:19_6"><i><font color="Green" title="E39">A39</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E14:19_6"/>
<font color="Maroon" title="c16">ss2</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a> 0
 <b>by </b><i><a class="txt" href="jordan20.html#E7:19"><i><font color="Green" title="E24">A5</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E11:19_6"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E15:19_6"/><b>then </b><i><font color="Green" title="E43">A43</font></i>: 
<font color="Maroon" title="c16">ss2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c6">f</font>
 <b>by </b><i><a class="txt" href="jordan20.html#E4:19"><i><font color="Green" title="E22">A3</font></i></a>, <a class="txt" href="jordan20.html#E8:19"><i><font color="Green" title="E25">A6</font></i></a>, <a class="txt" href="jordan20.html#E11:19_6"><i><font color="Green" title="E41">A41</font></i></a>, <a class="ref" href="rcomp_1.html#T48" title="RCOMP_1:th.48">RCOMP_1:48</a></i>;<br/><a NAME="E16:19_6"/>
<font color="Maroon" title="c6">f</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="jordan20.html#E3:19"><i><font color="Green" title="E21">A2</font></i></a>, <a class="ref" href="tops_2.html#D5" title="TOPS_2:def.5">TOPS_2:def 5</a></i>;<br/><b>hence </b><a NAME="E17:19_6"/>
contradiction
 <b>by </b><i><a class="txt" href="jordan20.html#E5:19_6"><i><font color="Green" title="E38">A38</font></i></a>, <a class="txt" href="jordan20.html#E7:19_6"><i><font color="Green" title="E39">A39</font></i></a>, <a class="txt" href="jordan20.html#E9:19_6"><i><font color="Green" title="E40">A40</font></i></a>, <a class="txt" href="jordan20.html#E11:19_6"><i><font color="Green" title="E41">A41</font></i></a>, <a class="txt" href="jordan20.html#E13:19_6"><i><font color="Green" title="E42">A42</font></i></a>, <a class="txt" href="jordan20.html#E15:19_6"><i><font color="Green" title="E43">A43</font></i></a>, <a class="ref" href="funct_1.html#D8" title="FUNCT_1:def.8">FUNCT_1:def 8</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E30:19"/>
not <font color="Maroon" title="c2">Q</font> is <a href="connsp_1.html#V2" title="CONNSP_1:attr.2">connected</a>
 <b>by </b><i><a class="txt" href="jordan20.html#E15:19"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="jordan20.html#E16:19"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="jordan20.html#E17:19"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="jordan20.html#E22:19"><i><font color="Green" title="E32">A27</font></i></a>, <a class="txt" href="jordan20.html#E27:19"><i><font color="Green" title="E35">A35</font></i></a>, <a class="ref" href="topreal5.html#T4" title="TOPREAL5:th.4">TOPREAL5:4</a></i>;<br/>

<b>reconsider </b><font color="Maroon" title="c12">T</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c11">Q'</font> as  non <a href="struct_0.html#V3" title="STRUCT_0:attr.3">empty</a> <a href="pre_topc.html#NM1" title="PRE_TOPC:NM.1">TopSpace</a> ;<br/>
<a NAME="E32:19"/><i><font color="Green" title="E37">A44</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c12">T</font>
 
;<br/>
<a NAME="E33:19"/><b>then </b><i><font color="Green" title="E38">A45</font></i>: 
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">Q</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>reconsider </b><font color="Maroon" title="c13">P1''</font> = <font color="Maroon" title="c9">P1'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">T</font> <b>by </b><i><a class="txt" href="jordan20.html#E20:19"><i><font color="Green" title="E31">A21</font></i></a>, <a class="txt" href="jordan20.html#E32:19"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E35:19"/>
<font color="Maroon" title="c13">P1''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">P1'</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E36:19"/><b>then </b><i><font color="Green" title="E39">A46</font></i>: 
<font color="Maroon" title="c13">P1''</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E15:19"><i><font color="Green" title="E27">A12</font></i></a>, <a class="txt" href="jordan20.html#E32:19"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">P2''</font> = <font color="Maroon" title="c10">P2'</font> as   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c12">T</font> <b>by </b><i><a class="txt" href="jordan20.html#E25:19"><i><font color="Green" title="E34">A29</font></i></a>, <a class="txt" href="jordan20.html#E32:19"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="pre_topc.html#D10" title="PRE_TOPC:def.10">PRE_TOPC:def 10</a></i>;<br/>
<a NAME="E38:19"/>
<font color="Maroon" title="c14">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">P2'</font> <a href="subset_1.html#K8" title="SUBSET_1:func.8">/\</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c12">T</font>
 
<b>by </b><i><a class="ref" href="xboole_1.html#T28" title="XBOOLE_1:th.28">XBOOLE_1:28</a></i>;<br/>
<a NAME="E39:19"/><b>then </b><i><font color="Green" title="E40">A47</font></i>: 
<font color="Maroon" title="c14">P2''</font> is <a href="pre_topc.html#V3" title="PRE_TOPC:attr.3">open</a>
 
<b>by </b><i><a class="txt" href="jordan20.html#E16:19"><i><font color="Green" title="E28">A13</font></i></a>, <a class="txt" href="jordan20.html#E32:19"><i><font color="Green" title="E37">A44</font></i></a>, <a class="ref" href="tops_2.html#T32" title="TOPS_2:th.32">TOPS_2:32</a></i>;<br/>
<a NAME="E40:19"/><i><font color="Green" title="E41">A48</font></i>: 
<font color="Maroon" title="c13">P1''</font> <a href="subset_1.html#K9" title="SUBSET_1:func.9">/\</a> <font color="Maroon" title="c14">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="xboole_0.html#K1" title="XBOOLE_0:func.1">{}</a> 
 
<b>by </b><i><a class="txt" href="jordan20.html#E29:19"><i><font color="Green" title="E36">A36</font></i></a>, <a class="ref" href="xboole_0.html#D7" title="XBOOLE_0:def.7">XBOOLE_0:def 7</a></i>;<br/>
<a NAME="E41:19"/>
<font color="Maroon" title="c13">P1''</font> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <font color="Maroon" title="c14">P2''</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>
 
<b>by </b><i><a class="txt" href="jordan20.html#E17:19"><i><font color="Green" title="E29">A14</font></i></a>, <a class="txt" href="jordan20.html#E33:19"><i><font color="Green" title="E38">A45</font></i></a>, <a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a></i>;<br/>
<b>hence </b><a NAME="E42:19"/>
for <font color="Olive" title="b1">R</font> being  <a href="struct_0.html#NM6" title="STRUCT_0:NM.6">Function</a> of <a href="topmetr.html#K5" title="TOPMETR:func.5">I[01]</a> ,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c1">P</font></span>)</span> <a href="pre_topc.html#K1" title="PRE_TOPC:func.1">|</a> <font color="Maroon" title="c2">Q</font></span>)</span>  holds <br/>(  not <font color="Olive" title="b1">R</font> is <a href="pre_topc.html#V5" title="PRE_TOPC:attr.5">continuous</a> or  not <font color="Olive" title="b1">R</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="c3">p1</font> or  not <font color="Olive" title="b1">R</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> 1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">p2</font> )
 <b>by </b><i><a class="txt" href="jordan20.html#E19:19"><i><font color="Green" title="E30">A20</font></i></a>, <a class="txt" href="jordan20.html#E24:19"><i><font color="Green" title="E33">A28</font></i></a>, <a class="txt" href="jordan20.html#E36:19"><i><font color="Green" title="E39">A46</font></i></a>, <a class="txt" href="jordan20.html#E39:19"><i><font color="Green" title="E40">A47</font></i></a>, <a class="txt" href="jordan20.html#E40:19"><i><font color="Green" title="E41">A48</font></i></a>, <a class="ref" href="jordan20.html#T17" target="_self" title="JORDAN20:th.17">Th17</a></i>;<br/>


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