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<b>let </b><font color="Maroon" title="c1">p</font>, <font color="Maroon" title="c2">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="E1:15"/><i><font color="Green" title="E8">A1</font></i>: 
<font color="Maroon" title="c1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> <font color="Maroon" title="c2">q</font>
 ;<br/>

<b>thus </b><a NAME="E2:15"/>
 <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> ) </span>}</span> 
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</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="E1:15_1"/><i><font color="Green" title="E9">A2</font></i>: 
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">a'</font> = <font color="Maroon" title="c3">a</font> as   <a href="pre_topc.html#NM2" title="PRE_TOPC:NM.2">Point</a> of <span class="p1">(<span class="default"><a href="euclid.html#K15" title="EUCLID:func.15">TOP-REAL</a> 2</span>)</span> ;<br/>
<a NAME="E3:15_1"/>
(  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> )
 
<b>by </b><i><a class="txt" href="jordan5b.html#E1:15"><i><font color="Green" title="E8">A1</font></i></a>, <a class="txt" href="jordan5b.html#E1:15_1"><i><font color="Green" title="E9">A2</font></i></a>, <a class="ref" href="jordan5b.html#T10" target="_self" title="JORDAN5B:th.10">Th10</a>, <a class="ref" href="jordan5b.html#T11" target="_self" title="JORDAN5B:th.11">Th11</a></i>;<br/>
<b>hence </b><a NAME="E4:15_1"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> ) </span>}</span> 
 ;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:15"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> ) </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="15_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">a</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="E1:15_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">p1</font> where <font color="Olive" title="b1">p1</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> : (  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Olive" title="b1">p1</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> ) </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c4">a'</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="E3:15_2"/><i><font color="Green" title="E9">A3</font></i>: 
( <font color="Maroon" title="c4">a'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c3">a</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> &amp;  <a href="jordan3.html#R2" title="JORDAN3:pred.2">LE</a> <font color="Maroon" title="c4">a'</font>,<font color="Maroon" title="c2">q</font>,<font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font> )
 ;<br/>
<b>thus </b><a NAME="E4:15_2"/>
<font color="Maroon" title="c3">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="topreal1.html#K3" title="TOPREAL1:func.3">LSeg</a> <font color="Maroon" title="c1">p</font>,<font color="Maroon" title="c2">q</font>
 <b>by </b><i><a class="txt" href="jordan5b.html#E3:15_2"><i><font color="Green" title="E9">A3</font></i></a>, <a class="ref" href="jordan3.html#T65" title="JORDAN3:th.65">JORDAN3:65</a></i>;<br/>


</div><b>end;</b></div>


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